Theorem rexlimiv 2662

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 1581	. 2 ⊢ Ⅎ𝑥𝜓
2		rexlimiv.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	rexlimi 2661	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209  ∃wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is used by:  rexlimiva  2663  rexlimivw  2664  rexlimivv  2674  r19.36av  2702  r19.44av  2710  r19.45av  2711  rexn0  3626  uniss2  3966  elres  5099  ssimaex  5764  mpoexw  6449  tfrlem5  6585  tfrlem8  6589  ecoptocl  6896  mapsn  6972  elixpsn  7017  ixpsnf1o  7018  findcard  7192  findcard2  7193  findcard2s  7194  fiintim  7238  prnmaddl  7857  0re  8326  cnegexlem2  8502  0cnALT  8516  bndndx  9562  uzn0  9938  ublbneg  10013  rexanuz2  11757  opnneiid  15265  2lgslem1b  16208  2sqlem2  16234  bj-inf2vnlem2  16997