Theorem cbveu 2069

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
cbveu.1	|- F/ y ph
cbveu.2	|- F/ x ps
cbveu.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbveu	|- ( E! x ph <-> E! y ps )

Proof of Theorem cbveu

Step	Hyp	Ref	Expression
1		cbveu.1	. . 3 |- F/ y ph
2	1	sb8eu 2058	. 2 |- ( E! x ph <-> E! y [ y / x ] ph )
3		cbveu.2	. . . 4 |- F/ x ps
4		cbveu.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
5	3, 4	sbie 1805	. . 3 |- ( [ y / x ] ph <-> ps )
6	5	eubii 2054	. 2 |- ( E! y [ y / x ] ph <-> E! y ps )
7	2, 6	bitri 184	1 |- ( E! x ph <-> E! y ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1474   [wsb 1776   E!weu 2045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048

This theorem is referenced by:  cbvmo  2085  cbvreu  2727  cbvreucsf  3149  tz6.12f  5587  f1ompt  5713  climeu  11461