Theorem rexlimiv 2543

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 1508	. 2 ⊢ Ⅎ𝑥𝜓
2		rexlimiv.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	rexlimi 2542	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 1480  ∃wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2421  df-rex 2422

This theorem is referenced by:  rexlimiva  2544  rexlimivw  2545  rexlimivv  2555  r19.36av  2582  r19.44av  2590  r19.45av  2591  rexn0  3461  uniss2  3767  elres  4855  ssimaex  5482  tfrlem5  6211  tfrlem8  6215  ecoptocl  6516  mapsn  6584  elixpsn  6629  ixpsnf1o  6630  findcard  6782  findcard2  6783  findcard2s  6784  fiintim  6817  prnmaddl  7298  0re  7766  cnegexlem2  7938  0cnALT  7952  bndndx  8976  uzn0  9341  ublbneg  9405  rexanuz2  10763  opnneiid  12333  bj-inf2vnlem2  13169