Theorem rexlimiv 2483

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 1466	. 2 ⊢ Ⅎ𝑥𝜓
2		rexlimiv.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	rexlimi 2482	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 1438  ∃wrex 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364  df-rex 2365

This theorem is referenced by:  rexlimiva  2484  rexlimivw  2485  rexlimivv  2494  r19.36av  2518  r19.44av  2526  r19.45av  2527  rexn0  3378  uniss2  3682  elres  4743  ssimaex  5359  tfrlem5  6071  tfrlem8  6075  ecoptocl  6369  mapsn  6437  findcard  6594  findcard2  6595  findcard2s  6596  fiintim  6629  prnmaddl  7039  0re  7478  cnegexlem2  7648  0cnALT  7662  bndndx  8662  uzn0  9024  ublbneg  9088  rexanuz2  10412  bj-inf2vnlem2  11749