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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem19.21t 1801 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
 |-  ( F/ x ph  ->  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
 
Theorem19.21 1802 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in  ph." (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
 
Theorem19.21h 1803 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in  ph." (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
 
Theoremstdpc5 1804 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis  F/ x ph can be thought of as emulating " x is not free in  ph." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example  x would not (for us) be free in  x  =  x by nfequid 1683. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  ->  ps )  ->  ( ph  ->  A. x ps )
 )
 
Theoremstdpc5OLD 1805 Obsolete proof of stdpc5 1804 as of 1-Jan-2018. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  ->  ps )  ->  ( ph  ->  A. x ps )
 )
 
Theorem19.23t 1806 Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
 |-  ( F/ x ps  ->  ( A. x (
 ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) ) )
 
Theorem19.23 1807 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ps   =>    |-  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) )
 
Theorem19.23h 1808 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
 |-  ( ps  ->  A. x ps )   =>    |-  ( A. x (
 ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) )
 
Theoremexlimi 1809 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ps   &    |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremexlimih 1810 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
 |-  ( ps  ->  A. x ps )   &    |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
TheoremexlimihOLD 1811 Obsolete proof of exlimih 1810 as of 1-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theoremexlimd 1812 Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
 |- 
 F/ x ph   &    |-  F/ x ch   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch )
 )
 
TheoremexlimdOLD 1813 Obsolete proof of exlimd 1812 as of 12-Jan-2018. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
 |- 
 F/ x ph   &    |-  F/ x ch   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch )
 )
 
Theoremexlimdh 1814 Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-Jan-1997.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch )
 )
 
Theoremnfimd 1815 If in a context  x is not free in  ps and  ch, it is not free in  ( ps  ->  ch ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  ->  ch ) )
 
TheoremnfimdOLD 1816 Obsolete proof of nfimd 1815 as of 29-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  ->  ch ) )
 
Theoremhbim1 1817 A closed form of hbim 1824. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ( ph  ->  ps )  ->  A. x (
 ph  ->  ps ) )
 
Theoremnfim1 1818 A closed form of nfim 1820. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  F/ x (
 ph  ->  ps )
 
Theoremnfim1OLD 1819 A closed form of nfim 1820. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  F/ x (
 ph  ->  ps )
 
Theoremnfim 1820 If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  ->  ps )
 
TheoremnfimOLD 1821 If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  ->  ps )
 
Theoremhbimd 1822 Deduction form of bound-variable hypothesis builder hbim 1824. (Contributed by NM, 1-Jan-2002.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   &    |-  ( ph  ->  ( ch  ->  A. x ch ) )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  A. x ( ps 
 ->  ch ) ) )
 
TheoremhbimdOLD 1823 Obsolete proof of hbimd 1822 as of 16-Dec-2017. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   &    |-  ( ph  ->  ( ch  ->  A. x ch ) )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  A. x ( ps 
 ->  ch ) ) )
 
Theoremhbim 1824 If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ( ph  ->  ps )  ->  A. x ( ph  ->  ps )
 )
 
TheoremhbimOLD 1825 Obsolete proof of hbim 1824 as of 1-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ( ph  ->  ps )  ->  A. x ( ph  ->  ps )
 )
 
Theorem19.23tOLD 1826 Obsolete proof of 19.23t 1806 as of 1-Jan-2018. (Contributed by NM, 7-Nov-2005.) (New usage is discouraged.)
 |-  ( F/ x ps  ->  ( A. x (
 ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) ) )
 
Theorem19.23hOLD 1827 Obsolete proof of 19.23h 1808 as of 1-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   =>    |-  ( A. x (
 ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) )
 
TheoremspimehOLD 1828* Obsolete proof of spimeh 1672 as of 10-Dec-2017. (Contributed by NM, 7-Aug-1994.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( x  =  z  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
Theorem19.27 1829 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( A. x ( ph  /\  ps )  <->  (
 A. x ph  /\  ps ) )
 
Theorem19.28 1830 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  /\  ps )  <->  (
 ph  /\  A. x ps ) )
 
Theoremnfand 1831 If in a context  x is not free in  ps and  ch, it is not free in  ( ps  /\  ch ). (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  /\  ch ) )
 
Theoremnf3and 1832 Deduction form of bound-variable hypothesis builder nf3an 1837. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   &    |-  ( ph  ->  F/ x th )   =>    |-  ( ph  ->  F/ x ( ps  /\  ch  /\  th ) )
 
Theoremnfan1 1833 A closed form of nfan 1834. (Contributed by Mario Carneiro, 3-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  F/ x (
 ph  /\  ps )
 
Theoremnfan 1834 If  x is not free in  ph and  ps, it is not free in  ( ph  /\  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  /\  ps )
 
Theoremnfnan 1835 If  x is not free in  ph and  ps, then it is not free in  ( ph  -/\  ps ). (Contributed by Scott Fenton, 2-Jan-2018.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  -/\  ps )
 
TheoremnfanOLD 1836 Obsolete proof of nfan 1834 as of 2-Jan-2018. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  /\  ps )
 
Theoremnf3an 1837 If  x is not free in  ph,  ps, and  ch, it is not free in  ( ph  /\  ps  /\  ch ). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   &    |-  F/ x ch   =>    |- 
 F/ x ( ph  /\ 
 ps  /\  ch )
 
Theoremhban 1838 If  x is not free in  ph and  ps, it is not free in  ( ph  /\  ps ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ( ph  /\ 
 ps )  ->  A. x ( ph  /\  ps )
 )
 
TheoremhbanOLD 1839 Obsolete proof of hban 1838 as of 2-Jan-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ( ph  /\ 
 ps )  ->  A. x ( ph  /\  ps )
 )
 
Theoremhb3an 1840 If  x is not free in  ph,  ps, and  ch, it is not free in  ( ph  /\  ps  /\  ch ). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ch  ->  A. x ch )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  A. x (
 ph  /\  ps  /\  ch ) )
 
Theoremhb3anOLD 1841 Obsolete proof of hb3an 1840 as of 2-Jan-2018. (Contributed by NM, 14-Sep-2003.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ch  ->  A. x ch )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  A. x (
 ph  /\  ps  /\  ch ) )
 
Theoremnfbid 1842 If in a context  x is not free in  ps and  ch, it is not free in  ( ps  <->  ch ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  <->  ch ) )
 
TheoremnfbidOLD 1843 Obsolete proof of nfbid 1842 as of 29-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  F/ x ( ps  <->  ch ) )
 
Theoremnfbi 1844 If  x is not free in  ph and  ps, it is not free in  ( ph  <->  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  <->  ps )
 
TheoremnfbiOLD 1845 If  x is not free in  ph and  ps, it is not free in  ( ph  <->  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  <->  ps )
 
Theoremnfor 1846 If  x is not free in  ph and  ps, it is not free in  ( ph  \/  ps ). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   =>    |-  F/ x ( ph  \/  ps )
 
Theoremnf3or 1847 If  x is not free in  ph,  ps, and  ch, it is not free in  ( ph  \/  ps  \/  ch ). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |-  F/ x ps   &    |-  F/ x ch   =>    |- 
 F/ x ( ph  \/  ps  \/  ch )
 
Theoremequsalhw 1848* Weaker version of equsalh 1974 (requiring distinct variables) without using ax-12 1937. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 28-Dec-2017.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
 
TheoremequsalhwOLD 1849* Obsolete proof of equsalhw 1848 as of 28-Dec-2017. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
 
Theorem19.21hOLD 1850 Obsolete proof of 19.21h 1803 as of 1-Jan-2018. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
 
Theoremhbex 1851 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( E. y ph  ->  A. x E. y ph )
 
Theoremnfal 1852 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x A. y ph
 
Theoremnfex 1853 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
 |- 
 F/ x ph   =>    |- 
 F/ x E. y ph
 
TheoremnfexOLD 1854 Obsolete proof of nfex 1853 as of 30-Dec-2017. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
 |- 
 F/ x ph   =>    |- 
 F/ x E. y ph
 
Theoremnfnf 1855 If  x is not free in  ph, it is not free in  F/ y ph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
 |- 
 F/ x ph   =>    |- 
 F/ x F/ y ph
 
TheoremnfnfOLD 1856 Obsolete proof of nfnf 1855 as of 30-Dec-2017. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
 |- 
 F/ x ph   =>    |- 
 F/ x F/ y ph
 
Theorem19.12 1857 Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 1908 and r19.12sn 3787. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
 |-  ( E. x A. y ph  ->  A. y E. x ph )
 
Theorem19.12OLD 1858 Obsolete proof of 19.12 1857 as of 3-Jan-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( E. x A. y ph  ->  A. y E. x ph )
 
Theoremdvelimhw 1859* Proof of dvelimh 1977 without using ax-12 1937 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   &    |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremcbv3hv 1860* Lemma for ax10 1957. Similar to cbv3h 1996. Requires distinct variables but avoids ax-12 1937. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremcbv3hvOLD 1861* Obsolete proof of cbv3hv 1860 as of 29-Dec-2017. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( ph  ->  A. y ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
Theoremnfald 1862 If  x is not free in  ph, it is not free in  A. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y ps )
 
TheoremnfaldOLD 1863 Obsolete proof of nfald 1862 as of 6-Jan-2018. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y ps )
 
Theoremnfexd 1864 If  x is not free in  ph, it is not free in  E. y ph. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y ps )
 
Theoremnfa2 1865 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x A. y A. x ph
 
Theoremnfia1 1866 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ( A. x ph  ->  A. x ps )
 
Theorem19.9tOLD 1867 Obsolete proof of 19.9t 1785 as of 30-Dec-2017. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
 |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
 
TheoremexcomimOLD 1868 Obsolete proof of excomim 1747 as of 8-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
 |-  ( E. x E. y ph  ->  E. y E. x ph )
 
TheoremexcomOLD 1869 Obsolete proof of excom 1746 as of 8-Jan-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( E. x E. y ph  <->  E. y E. x ph )
 
Theorem19.16 1870 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph 
 <->  ps )  ->  ( ph 
 <-> 
 A. x ps )
 )
 
Theorem19.17 1871 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( A. x ( ph  <->  ps )  ->  ( A. x ph  <->  ps ) )
 
Theorem19.19 1872 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph 
 <->  ps )  ->  ( ph 
 <-> 
 E. x ps )
 )
 
Theorem19.21tOLD 1873 Obsolete proof of 19.21t 1801 as of 30-Dec-2017. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.)
 |-  ( F/ x ph  ->  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
 
Theorem19.21-2 1874 Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
 |- 
 F/ x ph   &    |-  F/ y ph   =>    |-  ( A. x A. y (
 ph  ->  ps )  <->  ( ph  ->  A. x A. y ps ) )
 
Theorem19.21bbi 1875 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
 |-  ( ph  ->  A. x A. y ps )   =>    |-  ( ph  ->  ps )
 
Theoremnf2 1876 An alternative definition of df-nf 1550, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  <->  ( E. x ph  ->  A. x ph ) )
 
Theoremnf3 1877 An alternative definition of df-nf 1550. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  <->  A. x ( E. x ph 
 ->  ph ) )
 
Theoremnf4 1878 Variable  x is effectively not free in  ph iff  ph is always true or always false. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  <->  ( A. x ph  \/  A. x  -.  ph ) )
 
Theorem19.36 1879 Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( E. x ( ph  ->  ps )  <->  (
 A. x ph  ->  ps ) )
 
Theorem19.36i 1880 Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   &    |-  E. x ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theorem19.37 1881 Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( E. x (
 ph  ->  ps )  <->  ( ph  ->  E. x ps ) )
 
Theorem19.38OLD 1882 Obsolete proof of 19.38 1800 as of 2-Jan-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
 |-  ( ( E. x ph 
 ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
 
Theorem19.32 1883 Theorem 19.32 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  \/  ps )  <->  (
 ph  \/  A. x ps ) )
 
Theorem19.31 1884 Theorem 19.31 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( A. x ( ph  \/  ps )  <->  (
 A. x ph  \/  ps ) )
 
Theorem19.44 1885 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( E. x ( ph  \/  ps )  <->  ( E. x ph  \/  ps ) )
 
Theorem19.45 1886 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( E. x (
 ph  \/  ps )  <->  (
 ph  \/  E. x ps ) )
 
Theorem19.41 1887 Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
 |- 
 F/ x ps   =>    |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
 
Theorem19.41OLD 1888 Obsolete proof of 19.41 1887 as of 12-Jan-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
 |- 
 F/ x ps   =>    |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
 
Theorem19.42 1889 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( E. x (
 ph  /\  ps )  <->  (
 ph  /\  E. x ps ) )
 
Theoremexan 1890 Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
 |-  ( E. x ph  /\ 
 ps )   =>    |- 
 E. x ( ph  /\ 
 ps )
 
TheoremexanOLD 1891 Obsolete proof of exan 1890 as of 13-Jan-2018. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
 |-  ( E. x ph  /\ 
 ps )   =>    |- 
 E. x ( ph  /\ 
 ps )
 
Theoremhbnd 1892 Deduction form of bound-variable hypothesis builder hbn 1782. (Contributed by NM, 3-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  ( -.  ps  ->  A. x  -.  ps ) )
 
Theoremaaan 1893 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( A. x A. y (
 ph  /\  ps )  <->  (
 A. x ph  /\  A. y ps ) )
 
Theoremeeor 1894 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x E. y (
 ph  \/  ps )  <->  ( E. x ph  \/  E. y ps ) )
 
Theoremqexmid 1895 Quantified "excluded middle." Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)
 |- 
 E. x ( ph  ->  A. x ph )
 
Theoremequs5a 1896 A property related to substitution that unlike equs5 2009 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
 |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x ( x  =  y  ->  ph ) )
 
Theoremequs5e 1897 A property related to substitution that unlike equs5 2009 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.)
 |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
 
Theoremequs5eOLD 1898 Obsolete proof of equs5e 1897 as of 15-Jan-2018. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
 |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
 
Theoremexlimdd 1899 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |- 
 F/ x ph   &    |-  F/ x ch   &    |-  ( ph  ->  E. x ps )   &    |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theorem19.21v 1900* Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as  F/ x ph in 19.21 1802 via the use of distinct variable conditions combined with nfv 1624. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g. euf 2223 derived from df-eu 2221. The "f" stands for "not free in" which is less restrictive than "does not occur in." (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
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