Theorem cbvexdva 2283

Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2019. (Revised by Wolf Lammen, 18-Jul-2021.)

Hypothesis

Ref	Expression
cbvaldva.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
cbvexdva	|- ( ph -> ( E. x ps <-> E. y ch ) )

Distinct variable groups:   ps, y   ch, x   ph, x   ph, y

Allowed substitution hints:   ps( x)   ch( y)

Proof of Theorem cbvexdva

Step	Hyp	Ref	Expression
1		cbvaldva.1	. . . . 5 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2	1	notbid 308	. . . 4 |- ( ( ph /\ x = y ) -> ( -. ps <-> -. ch ) )
3	2	cbvaldva 2281	. . 3 |- ( ph -> ( A. x -. ps <-> A. y -. ch ) )
4		alnex 1706	. . 3 |- ( A. x -. ps <-> -. E. x ps )
5		alnex 1706	. . 3 |- ( A. y -. ch <-> -. E. y ch )
6	3, 4, 5	3bitr3g 302	. 2 |- ( ph -> ( -. E. x ps <-> -. E. y ch ) )
7	6	con4bid 307	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

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-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  cbvex2v  2287  cbvrexdva2  3176  isinf  8173