Theorem rexlimdvw 2654

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)

Hypothesis

Ref	Expression
rexlimdvw.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

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

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥

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

Proof of Theorem rexlimdvw

Step	Hyp	Ref	Expression
1		rexlimdvw.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	a1d 22	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
3	2	rexlimdv 2649	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∈ wcel 2202  ∃wrex 2511

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515  df-rex 2516

This theorem is referenced by:  nnpredcl  4721  qsss  6763  fodjuomnilemdc  7343  ltpopr  7815  ltsopr  7816  ltexprlemlol  7822  ltexprlemupu  7824  cauappcvgprlemrnd  7870  caucvgprlemrnd  7893  caucvgprprlemrnd  7921  suplocexprlemss  7935  suplocexprlemrl  7937  suplocsrlempr  8027  climuni  11858  ellspsn  14437  cncnp2m  14961  bj-findis  16600