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Theorem equcoms 1825
Description: An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
equcoms.1  |-  ( x  =  y  ->  ph )
Assertion
Ref Expression
equcoms  |-  ( y  =  x  ->  ph )

Proof of Theorem equcoms
StepHypRef Expression
1 equcomi 1822 . 2  |-  ( y  =  x  ->  x  =  y )
2 equcoms.1 . 2  |-  ( x  =  y  ->  ph )
31, 2syl 17 1  |-  ( y  =  x  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6
This theorem is referenced by:  equtr  1826  equtr2  1828  equequ2  1830  elequ1  1831  elequ2  1832  ax10o  1834  cbval  1876  equvini  1879  ax12  1881  stdpc7  1891  sbequ12r  1893  sbequ12a  1894  sbequ  1952  sb6rf  1985  sb6a  2076  mo  2135  cleqh  2346  cbvab  2367  reu8  2900  tfinds2  4545  boxriin  6744  elirrv  7195  elmptrab  17354  pcoass  18354  cvmsss2  22976  ax13dfeq  23323  dominc  24446  rninc  24447  pdiveql  25334  sdclem2  25618  rexzrexnn0  26051  bnj1014  27681
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-8 1623  ax-17 1628  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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