Theorem rexlimdva 2409

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)

Hypothesis

Ref	Expression
rexlimdva.1	⊢ ((φ ∧ x ∈ A) → (ψ → χ))

Assertion

Ref	Expression
rexlimdva	⊢ (φ → (∃x ∈ A ψ → χ))

Distinct variable groups:   φ,x   χ,x

Allowed substitution hints:   ψ(x)   A(x)

Proof of Theorem rexlimdva

Step	Hyp	Ref	Expression
1		rexlimdva.1	. . 3 ⊢ ((φ ∧ x ∈ A) → (ψ → χ))
2	1	ex 108	. 2 ⊢ (φ → (x ∈ A → (ψ → χ)))
3	2	rexlimdv 2408	1 ⊢ (φ → (∃x ∈ A ψ → χ))

Syntax hints:   → wi 4   ∧ wa 97   ∈ 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:  rexlimdvaa  2410  rexlimivv  2414  rexlimdvv  2415  ralxfrd  4142  rexxfrd  4143  fvelimab  5152  foco2  5241  elunirn  5328  f1elima  5335  tfrlem5  5850  tfrlemibacc  5859  tfrlemibfn  5861  nnaordex  6009  nnawordex  6010  ectocld  6081  ltexnqq  6264  ltbtwnnqq  6270  prarloclem4  6350  prarloc2  6356  genprndl  6374  genprndu  6375  prmuloc2  6409  1idprl  6427  1idpru  6428  recexsrlem  6518