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  3964  elres  5097  ssimaex  5761  mpoexw  6443  tfrlem5  6579  tfrlem8  6583  ecoptocl  6890  mapsn  6966  elixpsn  7011  ixpsnf1o  7012  findcard  7186  findcard2  7187  findcard2s  7188  fiintim  7232  prnmaddl  7851  0re  8320  cnegexlem2  8496  0cnALT  8510  bndndx  9545  uzn0  9921  ublbneg  9996  rexanuz2  11740  opnneiid  15248  2lgslem1b  16191  2sqlem2  16217  bj-inf2vnlem2  16980