Theorem rexlimdva2 2617

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 2613	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   E.wrex 2476

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-ral 2480  df-rex 2481

This theorem is referenced by:  ctssdclemn0  7185  ctssdc  7188  suplocexprlemru  7805  suplocexprlemloc  7807  suplocsrlemb  7892  aptap  8696  4sqlemffi  12592  4sqleminfi  12593  4sqexercise2  12595  4sqlemsdc  12596  ennnfonelemhom  12659  gsumfzval  13095  reldvdsrsrg  13726  innei  14507  ivthinclemlr  14981  ivthinclemur  14983  limccnpcntop  15019  limccoap  15022  2lgslem1c  15439  2lgslem3a1  15446  2lgslem3b1  15447  2lgslem3c1  15448  2lgslem3d1  15449