Theorem rexlimivw 2405

Description: Weaker version of rexlimiv 2403. (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 2403	1 ⊢ (∃x ∈ A φ → ψ)

Syntax hints:   → wi 4   ∈ wcel 1374  ∃wrex 2283

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-i5r 1410

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-ral 2287  df-rex 2288

This theorem is referenced by:  r19.29_2a  2432  eliun  3633  reusv3i  4139  elrnmptg  4511  fun11iun  5070  fmpt  5242  fliftfun  5359  elrnmpt2  5535  releldm2  5732  tfrlem4  5849  iinerm  6087  ltbtwnnqq  6270  recexprlemlol  6458  recexprlemupu  6460