Theorem rexlimiva 2544

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)

Hypothesis

Ref	Expression
rexlimiva.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)

Assertion

Ref	Expression
rexlimiva	⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimiva

Step	Hyp	Ref	Expression
1		rexlimiva.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)
2	1	ex 114	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	rexlimiv 2543	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ∈ 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:  unon  4427  reg2exmidlema  4449  ssfilem  6769  diffitest  6781  fival  6858  elfi2  6860  fi0  6863  djuss  6955  updjud  6967  enumct  7000  finnum  7039  dmaddpqlem  7192  nqpi  7193  nq0nn  7257  recexprlemm  7439  rexanuz  10767  r19.2uz  10772  maxleast  10992  fsum2dlemstep  11210  fisumcom2  11214  0dvds  11520  even2n  11578  m1expe  11603  m1exp1  11605  epttop  12269  neipsm  12333  tgioo  12725  sin0pilem2  12873  pilem3  12874  bj-nn0suc  13172  bj-nn0sucALT  13186