Theorem equsex 2038

Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.)

Hypotheses

Ref	Expression
equsex.1
equsex.2

Assertion

Ref	Expression
equsex

Proof of Theorem equsex

Step	Hyp	Ref	Expression
1		equsex.1	. . 3
2		equsex.2	. . . 4
3	2	biimpa 484	. . 3
4	1, 3	exlimi 1912	. 2
5	1, 2	equsal 2036	. . 3
6		equs4 2035	. . 3
7	5, 6	sylbir 213	. 2
8	4, 7	impbii 188	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  F/wnf 1616

This theorem is referenced by:  equsexh  2039  cleljustALT  2110  sb5rf  2165  sb56  2172  sb10f  2200  axsep  4572  dprd2d2  17093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-12 1854  ax-13 1999

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617