Theorem rexlimivw 2543

Description: Weaker version of rexlimiv 2541. (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 2541	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 1480  ∃wrex 2415

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 2419  df-rex 2420

This theorem is referenced by:  r19.29vva  2575  eliun  3812  reusv3i  4375  elrnmptg  4786  fun11iun  5381  fmpt  5563  fliftfun  5690  elrnmpo  5877  releldm2  6076  tfrlem4  6203  iinerm  6494  elixpsn  6622  isfi  6648  cardcl  7030  cardval3ex  7034  ltbtwnnqq  7216  recexprlemlol  7427  recexprlemupu  7429  suplocsr  7610  restsspw  12119  ssnei  12309  tgcnp  12367  xmetunirn  12516  metss  12652  metrest  12664