Theorem rexlimd 2580

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 2537	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4		rexlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	r19.23 2574	. 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 1448   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449  df-rex 2450

This theorem is referenced by:  rexlimdv  2582  ralxfrALT  4445  fvmptt  5577  ffnfv  5643  nneneq  6823  ac6sfi  6864  prarloclem3step  7437  prmuloc2  7508  caucvgprprlemaddq  7649  axpre-suploclemres  7842  lbzbi  9554  divalglemeunn  11858  divalglemeuneg  11860  oddpwdclemdvds  12102  oddpwdclemndvds  12103  trirec0  13923