Theorem rexlimiv 2642

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 1574	. 2 ⊢ Ⅎ𝑥𝜓
2		rexlimiv.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	rexlimi 2641	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2200  ∃wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  rexlimiva  2643  rexlimivw  2644  rexlimivv  2654  r19.36av  2682  r19.44av  2690  r19.45av  2691  rexn0  3590  uniss2  3919  elres  5041  ssimaex  5697  mpoexw  6365  tfrlem5  6466  tfrlem8  6470  ecoptocl  6777  mapsn  6845  elixpsn  6890  ixpsnf1o  6891  findcard  7058  findcard2  7059  findcard2s  7060  fiintim  7104  prnmaddl  7688  0re  8157  cnegexlem2  8333  0cnALT  8347  bndndx  9379  uzn0  9750  ublbneg  9820  rexanuz2  11517  opnneiid  14853  2lgslem1b  15783  2sqlem2  15809  bj-inf2vnlem2  16389