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  3918  elres  5040  ssimaex  5694  mpoexw  6357  tfrlem5  6458  tfrlem8  6462  ecoptocl  6767  mapsn  6835  elixpsn  6880  ixpsnf1o  6881  findcard  7046  findcard2  7047  findcard2s  7048  fiintim  7089  prnmaddl  7673  0re  8142  cnegexlem2  8318  0cnALT  8332  bndndx  9364  uzn0  9734  ublbneg  9804  rexanuz2  11497  opnneiid  14832  2lgslem1b  15762  2sqlem2  15788  bj-inf2vnlem2  16292