Theorem rexlimiv 2605

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)

Hypothesis

Ref	Expression
rexlimiv.1	⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))

Assertion

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

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimiv

Step	Hyp	Ref	Expression
1		nfv 1539	. 2 ⊢ Ⅎ𝑥𝜓
2		rexlimiv.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	rexlimi 2604	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2164  ∃wrex 2473

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2477  df-rex 2478

This theorem is referenced by:  rexlimiva  2606  rexlimivw  2607  rexlimivv  2617  r19.36av  2645  r19.44av  2653  r19.45av  2654  rexn0  3545  uniss2  3866  elres  4978  ssimaex  5618  mpoexw  6266  tfrlem5  6367  tfrlem8  6371  ecoptocl  6676  mapsn  6744  elixpsn  6789  ixpsnf1o  6790  findcard  6944  findcard2  6945  findcard2s  6946  fiintim  6985  prnmaddl  7550  0re  8019  cnegexlem2  8195  0cnALT  8209  bndndx  9239  uzn0  9608  ublbneg  9678  rexanuz2  11135  opnneiid  14332  2sqlem2  15202  bj-inf2vnlem2  15463