Theorem cbveu 2505

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 2503	. 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 2408	. . 3 |- ( [ y / x ] ph <-> ps )
6	5	eubii 2492	. 2 |- ( E! y [ y / x ] ph <-> E! y ps )
7	2, 6	bitri 264	1 |- ( E! x ph <-> E! y ps )

Syntax hints:   -> wi 4   <-> wb 196   F/wnf 1708   [wsb 1880   E!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474

This theorem is referenced by:  cbvmo  2506  cbvreu  3169  cbvreucsf  3567  tz6.12f  6212  f1ompt  6382  climeu  14286  initoeu2  16666