Theorem rexlimdva 2427

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 2426	1 ⊢ (φ → (∃x ∈ A ψ → χ))

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  ∃wrex 2301

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305  df-rex 2306

This theorem is referenced by:  rexlimdvaa  2428  rexlimivv  2432  rexlimdvv  2433  ralxfrd  4160  rexxfrd  4161  fvelimab  5172  foco2  5261  elunirn  5348  f1elima  5355  tfrlem5  5871  tfrlemibacc  5881  tfrlemibfn  5883  nnaordex  6036  nnawordex  6037  ectocld  6108  ltexnqq  6391  ltbtwnnqq  6398  prarloclem4  6481  prarloc2  6487  genprndl  6504  genprndu  6505  prmuloc2  6548  1idprl  6566  1idpru  6567  cauappcvgprlemdisj  6623  cauappcvgprlemladdru  6628  cauappcvgprlemladdrl  6629  caucvgprlemladdrl  6649  recexgt0sr  6701  cnegexlem1  6983  cnegexlem2  6984  renegcl  7068  qmulz  8334  icc0r  8565  frec2uzrand  8872  frecuzrdgfn  8879