Theorem rexlimivw 2504

Description: Weaker version of rexlimiv 2502. (Contributed by FL, 19-Sep-2011.)

Hypothesis

Ref	Expression
rexlimivw.1	⊢ (𝜑 → 𝜓)

Assertion

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

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem rexlimivw

Step	Hyp	Ref	Expression
1		rexlimivw.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	a1i 9	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	rexlimiv 2502	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 1448  ∃wrex 2376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-17 1474  ax-ial 1482  ax-i5r 1483

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-ral 2380  df-rex 2381

This theorem is referenced by:  r19.29vva  2535  eliun  3764  reusv3i  4318  elrnmptg  4729  fun11iun  5322  fmpt  5502  fliftfun  5629  elrnmpo  5816  releldm2  6013  tfrlem4  6140  iinerm  6431  elixpsn  6559  isfi  6585  cardcl  6948  cardval3ex  6952  ltbtwnnqq  7124  recexprlemlol  7335  recexprlemupu  7337  restsspw  11912  ssnei  12102  tgcnp  12159  xmetunirn  12286  metss  12422  metrest  12434