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  3969  reusv3i  4550  elrnmptg  4976  fun11iun  5595  fmpt  5787  fliftfun  5926  elrnmpo  6124  releldm2  6337  tfrlem4  6465  iinerm  6762  elixpsn  6890  isfi  6920  cardcl  7364  cardval3ex  7368  ltbtwnnqq  7613  recexprlemlol  7824  recexprlemupu  7826  suplocsr  8007  restsspw  13297  rhmdvdsr  14154  ssnei  14840  tgcnp  14898  xmetunirn  15047  metss  15183  metrest  15195