Theorem rexlimdvv 2433

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)

Hypothesis

Ref	Expression
rexlimdvv.1	⊢ (φ → ((x ∈ A ∧ y ∈ B) → (ψ → χ)))

Assertion

Ref	Expression
rexlimdvv	⊢ (φ → (∃x ∈ A ∃y ∈ B ψ → χ))

Distinct variable groups:   x,y,φ   χ,x,y   y,A

Allowed substitution hints:   ψ(x,y)   A(x)   B(x,y)

Proof of Theorem rexlimdvv

Step	Hyp	Ref	Expression
1		rexlimdvv.1	. . . 4 ⊢ (φ → ((x ∈ A ∧ y ∈ B) → (ψ → χ)))
2	1	expdimp 246	. . 3 ⊢ ((φ ∧ x ∈ A) → (y ∈ B → (ψ → χ)))
3	2	rexlimdv 2426	. 2 ⊢ ((φ ∧ x ∈ A) → (∃y ∈ B ψ → χ))
4	3	rexlimdva 2427	1 ⊢ (φ → (∃x ∈ A ∃y ∈ B 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:  rexlimdvva  2434  f1oiso2  5409  xpdom2  6241  genpcdl  6502  genpcuu  6503  distrlem1prl  6558  distrlem1pru  6559  distrlem5prl  6562  distrlem5pru  6563  recexprlemss1l  6607  recexprlemss1u  6608  qaddcl  8346  qmulcl  8348