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  3972  reusv3i  4554  elrnmptg  4982  fun11iun  5601  fmpt  5793  fliftfun  5932  elrnmpo  6130  releldm2  6343  tfrlem4  6474  iinerm  6771  elixpsn  6899  isfi  6929  cardcl  7376  cardval3ex  7380  ltbtwnnqq  7625  recexprlemlol  7836  recexprlemupu  7838  suplocsr  8019  restsspw  13322  rhmdvdsr  14179  ssnei  14865  tgcnp  14923  xmetunirn  15072  metss  15208  metrest  15220  clwwlknun  16236