Theorem rexlimd 2659

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 2615	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4		rexlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	r19.23 2653	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
6	3, 5	sylib 122	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4  Ⅎwnf 1509   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523

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-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2527  df-rex 2528

This theorem is referenced by:  rexlimdv  2661  ralxfrALT  4593  fvmptt  5774  ffnfv  5840  elabreximd  6329  nneneq  7124  ac6sfi  7168  prarloclem3step  7827  prmuloc2  7898  caucvgprprlemaddq  8039  axpre-suploclemres  8232  lbzbi  9966  reuccatpfxs1  11464  divalglemeunn  12632  divalglemeuneg  12634  oddpwdclemdvds  12892  oddpwdclemndvds  12893  trirec0  16954