Theorem rexlimivw 2590

Description: Weaker version of rexlimiv 2588. (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 2588	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2148  ∃wrex 2456

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460  df-rex 2461

This theorem is referenced by:  r19.29vva  2622  eliun  3892  reusv3i  4461  elrnmptg  4881  fun11iun  5484  fmpt  5669  fliftfun  5800  elrnmpo  5991  releldm2  6189  tfrlem4  6317  iinerm  6610  elixpsn  6738  isfi  6764  cardcl  7183  cardval3ex  7187  ltbtwnnqq  7417  recexprlemlol  7628  recexprlemupu  7630  suplocsr  7811  restsspw  12704  ssnei  13791  tgcnp  13849  xmetunirn  13998  metss  14134  metrest  14146