Theorem rexlimivw 2644

Description: Weaker version of rexlimiv 2642. (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 2642	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2200  ∃wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  r19.29vva  2676  eliun  3968  reusv3i  4549  elrnmptg  4975  fun11iun  5592  fmpt  5784  fliftfun  5919  elrnmpo  6117  releldm2  6329  tfrlem4  6457  iinerm  6752  elixpsn  6880  isfi  6910  cardcl  7349  cardval3ex  7353  ltbtwnnqq  7598  recexprlemlol  7809  recexprlemupu  7811  suplocsr  7992  restsspw  13277  rhmdvdsr  14133  ssnei  14819  tgcnp  14877  xmetunirn  15026  metss  15162  metrest  15174