Theorem rexlimd 2549

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 2506	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4		rexlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	r19.23 2543	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
6	3, 5	sylib 121	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4  Ⅎwnf 1437   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-ral 2422  df-rex 2423

This theorem is referenced by:  rexlimdv  2551  ralxfrALT  4396  fvmptt  5520  ffnfv  5586  nneneq  6759  ac6sfi  6800  prarloclem3step  7328  prmuloc2  7399  caucvgprprlemaddq  7540  axpre-suploclemres  7733  lbzbi  9435  divalglemeunn  11654  divalglemeuneg  11656  oddpwdclemdvds  11884  oddpwdclemndvds  11885  trirec0  13412