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  6247  tfrlem4  6375  iinerm  6670  elixpsn  6798  isfi  6824  cardcl  7252  cardval3ex  7256  ltbtwnnqq  7487  recexprlemlol  7698  recexprlemupu  7700  suplocsr  7881  restsspw  12939  rhmdvdsr  13778  ssnei  14434  tgcnp  14492  xmetunirn  14641  metss  14777  metrest  14789