Theorem rexlimdva2 2614

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
rexlimdva2.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )

Assertion

Ref	Expression
rexlimdva2	|- ( ph -> ( E. x e. A ps -> ch ) )

Distinct variable groups:   ch, x   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem rexlimdva2

Step	Hyp	Ref	Expression
1		rexlimdva2.1	. . 3 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
2	1	exp31 364	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	2	rexlimdv 2610	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2164   E.wrex 2473

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-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2477  df-rex 2478

This theorem is referenced by:  ctssdclemn0  7169  ctssdc  7172  suplocexprlemru  7779  suplocexprlemloc  7781  suplocsrlemb  7866  aptap  8669  4sqlemffi  12534  4sqleminfi  12535  4sqexercise2  12537  4sqlemsdc  12538  ennnfonelemhom  12572  gsumfzval  12974  reldvdsrsrg  13588  innei  14331  ivthinclemlr  14791  ivthinclemur  14793  limccnpcntop  14829  limccoap  14832