Theorem rexlimivw 2619

Description: Weaker version of rexlimiv 2617. (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 2617	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2176  ∃wrex 2485

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-ral 2489  df-rex 2490

This theorem is referenced by:  r19.29vva  2651  eliun  3931  reusv3i  4506  elrnmptg  4930  fun11iun  5543  fmpt  5730  fliftfun  5865  elrnmpo  6059  releldm2  6271  tfrlem4  6399  iinerm  6694  elixpsn  6822  isfi  6852  cardcl  7288  cardval3ex  7292  ltbtwnnqq  7528  recexprlemlol  7739  recexprlemupu  7741  suplocsr  7922  restsspw  13081  rhmdvdsr  13937  ssnei  14623  tgcnp  14681  xmetunirn  14830  metss  14966  metrest  14978