Theorem rexlimivw 2658

Description: Weaker version of rexlimiv 2656. (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 2656	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2205  ∃wrex 2523

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2527  df-rex 2528

This theorem is referenced by:  r19.29vva  2690  eliun  4000  reusv3i  4585  elrnmptg  5014  fun11iun  5640  fmpt  5832  fliftfun  5975  elrnmpo  6175  releldm2  6392  tfrlem4  6557  iinerm  6854  elixpsn  6983  isfi  7013  cardcl  7490  cardval3ex  7494  ltbtwnnqq  7746  recexprlemlol  7957  recexprlemupu  7959  suplocsr  8140  restsspw  13546  rhmdvdsr  14420  ssnei  15142  tgcnp  15200  xmetunirn  15349  metss  15485  metrest  15497  clwwlknun  16562