Theorem reximdai 2595

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)

Hypotheses

Ref	Expression
reximdai.1	⊢ Ⅎ𝑥𝜑
reximdai.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))

Assertion

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

Proof of Theorem reximdai

Step	Hyp	Ref	Expression
1		reximdai.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		reximdai.2	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
3	1, 2	ralrimi 2568	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4		rexim 2591	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
5	3, 4	syl 14	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4  Ⅎwnf 1474   ∈ wcel 2167  ∀wral 2475  ∃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-ial 1548

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

This theorem is referenced by:  reximdvai  2597  bezoutlemstep  12164  isomninnlem  15674  ismkvnnlem  15696