Theorem equvin 1804

Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 1837, ax-13 1999. (Revised by Wolf Lammen, 10-Jun-2019.)

Assertion

Ref	Expression
equvin

Distinct variable groups:   ,   ,

Proof of Theorem equvin

Step	Hyp	Ref	Expression
1		equviniv 1803	. . 3
2		equcom 1794	. . . . 5
3	2	anbi2i 694	. . . 4
4	3	exbii 1667	. . 3
5	1, 4	sylib 196	. 2
6		equtr 1796	. . . 4
7	6	imp 429	. . 3
8	7	exlimiv 1722	. 2
9	5, 8	impbii 188	1

Syntax hints:  <->wb 184  /\wa 369  E.wex 1612

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

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