Theorem rexlimivw 2610

Description: Weaker version of rexlimiv 2608. (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 2608	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2167  ∃wrex 2476

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-ral 2480  df-rex 2481

This theorem is referenced by:  r19.29vva  2642  eliun  3921  reusv3i  4495  elrnmptg  4919  fun11iun  5528  fmpt  5715  fliftfun  5846  elrnmpo  6040  releldm2  6252  tfrlem4  6380  iinerm  6675  elixpsn  6803  isfi  6829  cardcl  7259  cardval3ex  7263  ltbtwnnqq  7499  recexprlemlol  7710  recexprlemupu  7712  suplocsr  7893  restsspw  12951  rhmdvdsr  13807  ssnei  14471  tgcnp  14529  xmetunirn  14678  metss  14814  metrest  14826