Theorem rexlimivw 2656

Description: Weaker version of rexlimiv 2654. (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 2654	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2203  ∃wrex 2521

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 2525  df-rex 2526

This theorem is referenced by:  r19.29vva  2688  eliun  3995  reusv3i  4580  elrnmptg  5009  fun11iun  5635  fmpt  5827  fliftfun  5969  elrnmpo  6167  releldm2  6379  tfrlem4  6544  iinerm  6841  elixpsn  6970  isfi  7000  cardcl  7477  cardval3ex  7481  ltbtwnnqq  7730  recexprlemlol  7941  recexprlemupu  7943  suplocsr  8124  restsspw  13462  rhmdvdsr  14320  ssnei  15016  tgcnp  15074  xmetunirn  15223  metss  15359  metrest  15371  clwwlknun  16436