Theorem rexlimdva2 2663

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 2659	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2203  ∃wrex 2521

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2525  df-rex 2526

This theorem is referenced by:  ctssdclemn0  7400  ctssdc  7403  suplocexprlemru  8033  suplocexprlemloc  8035  suplocsrlemb  8120  aptap  8923  4sqlemffi  13090  4sqleminfi  13091  4sqexercise2  13093  4sqlemsdc  13094  ennnfonelemhom  13158  gsumfzval  13596  innei  15020  ivthinclemlr  15494  ivthinclemur  15496  limccnpcntop  15532  limccoap  15535  2lgslem1c  15955  2lgslem3a1  15962  2lgslem3b1  15963  2lgslem3c1  15964  2lgslem3d1  15965  umgrnloop  16103