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  3591  uniss2  3922  elres  5047  ssimaex  5703  mpoexw  6373  tfrlem5  6475  tfrlem8  6479  ecoptocl  6786  mapsn  6854  elixpsn  6899  ixpsnf1o  6900  findcard  7070  findcard2  7071  findcard2s  7072  fiintim  7116  prnmaddl  7700  0re  8169  cnegexlem2  8345  0cnALT  8359  bndndx  9391  uzn0  9762  ublbneg  9837  rexanuz2  11542  opnneiid  14878  2lgslem1b  15808  2sqlem2  15834  bj-inf2vnlem2  16502