Theorem cbveu 1965

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 1954	. 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 1714	. . 3 |- ( [ y / x ] ph <-> ps )
6	5	eubii 1950	. 2 |- ( E! y [ y / x ] ph <-> E! y ps )
7	2, 6	bitri 182	1 |- ( E! x ph <-> E! y ps )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1389   [wsb 1685   E!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944

This theorem is referenced by:  cbvmo  1981  cbvreu  2575  cbvreucsf  2966  tz6.12f  5223  f1ompt  5341  climeu  10135