Theorem 19.21t 1795

Description: 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.)

Assertion

Ref	Expression
19.21t	|- ( F/xph -> (A.x(ph -> ps) <-> (ph -> A.xps)))

Proof of Theorem 19.21t

Step	Hyp	Ref	Expression
1		nfr 1761	. . 3 |- ( F/xph -> (ph -> A.xph))
2		ax-5 1557	. . 3 |- (A.x(ph -> ps) -> (A.xph -> A.xps))
3	1, 2	syl9 66	. 2 |- ( F/xph -> (A.x(ph -> ps) -> (ph -> A.xps)))
4		19.9t 1779	. . . 4 |- ( F/xph -> (E.xph <-> ph))
5	4	imbi1d 308	. . 3 |- ( F/xph -> ((E.xph -> A.xps) <-> (ph -> A.xps)))
6		19.38 1794	. . 3 |- ((E.xph -> A.xps) -> A.x(ph -> ps))
7	5, 6	syl6bir 220	. 2 |- ( F/xph -> ((ph -> A.xps) -> A.x(ph -> ps)))
8	3, 7	impbid 183	1 |- ( F/xph -> (A.x(ph -> ps) <-> (ph -> A.xps)))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  19.21  1796  nfimd  1808  sbcom  2089  sbal2  2134  ax11indalem  2197  ax11inda2ALT  2198  r19.21t  2700  ceqsalt  2882  sbciegft  3077