Theorem rexlimiv 2644

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 1576	. 2 ⊢ Ⅎ𝑥𝜓
2		rexlimiv.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	rexlimi 2643	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2202  ∃wrex 2511

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515  df-rex 2516

This theorem is referenced by:  rexlimiva  2645  rexlimivw  2646  rexlimivv  2656  r19.36av  2684  r19.44av  2692  r19.45av  2693  rexn0  3593  uniss2  3924  elres  5049  ssimaex  5707  mpoexw  6378  tfrlem5  6480  tfrlem8  6484  ecoptocl  6791  mapsn  6859  elixpsn  6904  ixpsnf1o  6905  findcard  7077  findcard2  7078  findcard2s  7079  fiintim  7123  prnmaddl  7710  0re  8179  cnegexlem2  8355  0cnALT  8369  bndndx  9401  uzn0  9772  ublbneg  9847  rexanuz2  11556  opnneiid  14894  2lgslem1b  15824  2sqlem2  15850  bj-inf2vnlem2  16592