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)

Syntax hints:   → wi 4   ∈ wcel 2209  ∃wrex 2529

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

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

This theorem is referenced by:  rexlimiva  2663  rexlimivw  2664  rexlimivv  2674  r19.36av  2702  r19.44av  2710  r19.45av  2711  rexn0  3623  uniss2  3961  elres  5094  ssimaex  5758  mpoexw  6439  tfrlem5  6575  tfrlem8  6579  ecoptocl  6886  mapsn  6962  elixpsn  7007  ixpsnf1o  7008  findcard  7182  findcard2  7183  findcard2s  7184  fiintim  7228  prnmaddl  7847  0re  8316  cnegexlem2  8492  0cnALT  8506  bndndx  9541  uzn0  9917  ublbneg  9992  rexanuz2  11735  opnneiid  15188  2lgslem1b  16122  2sqlem2  16148  bj-inf2vnlem2  16911