Theorem rexlimdvw 2652

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 2647	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∈ wcel 2200  ∃wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  nnpredcl  4719  qsss  6758  fodjuomnilemdc  7334  ltpopr  7805  ltsopr  7806  ltexprlemlol  7812  ltexprlemupu  7814  cauappcvgprlemrnd  7860  caucvgprlemrnd  7883  caucvgprprlemrnd  7911  suplocexprlemss  7925  suplocexprlemrl  7927  suplocsrlempr  8017  climuni  11844  ellspsn  14421  cncnp2m  14945  bj-findis  16510