Theorem rexlimdva2 2671

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	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
rexlimdva2	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Distinct variable groups:   𝜒,𝑥   𝜑,𝑥

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

Proof of Theorem rexlimdva2

Step	Hyp	Ref	Expression
1		rexlimdva2.1	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
2	1	exp31 364	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
3	2	rexlimdv 2667	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2209  ∃wrex 2529

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is referenced by:  ctssdclemn0  7444  ctssdc  7447  suplocexprlemru  8080  suplocexprlemloc  8082  suplocsrlemb  8167  aptap  8972  4sqlemffi  13158  4sqleminfi  13159  4sqexercise2  13161  4sqlemsdc  13162  ennnfonelemhom  13289  gzsumfzval  13694  innei  15247  ivthinclemlr  15721  ivthinclemur  15723  limccnpcntop  15759  limccoap  15762  2lgslem1c  16192  2lgslem3a1  16199  2lgslem3b1  16200  2lgslem3c1  16201  2lgslem3d1  16202  umgrnloop  16340