Theorem cbveu 2024

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 2013	. 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 1765	. . 3 |- ( [ y / x ] ph <-> ps )
6	5	eubii 2009	. 2 |- ( E! y [ y / x ] ph <-> E! y ps )
7	2, 6	bitri 183	1 |- ( E! x ph <-> E! y ps )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1437   [wsb 1736   E!weu 2000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003

This theorem is referenced by:  cbvmo  2040  cbvreu  2655  cbvreucsf  3069  tz6.12f  5458  f1ompt  5579  climeu  11097