Theorem rexlimd 2430

Description: Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

Ref	Expression
rexlimd.1	⊢ Ⅎ𝑥𝜑
rexlimd.2	⊢ Ⅎ𝑥𝜒
rexlimd.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))

Assertion

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

Proof of Theorem rexlimd

Step	Hyp	Ref	Expression
1		rexlimd.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rexlimd.3	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
3	1, 2	ralrimi 2390	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4		rexlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	r19.23 2424	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
6	3, 5	sylib 127	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4  Ⅎwnf 1349   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by:  rexlimdv  2432  ralxfrALT  4199  fvmptt  5262  ffnfv  5323  nneneq  6320  ac6sfi  6352  prarloclem3step  6594  prmuloc2  6665  caucvgprprlemaddq  6806  lbzbi  8551