Theorem r19.23advv 1288

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.)

Hypothesis

Ref	Expression
r19.23advv.1	⊢ (φ → ((x ∈ A ∧ y ∈ B) → (ψ → χ)))

Assertion

Ref	Expression
r19.23advv	⊢ (φ → (∃x ∈ A ∃y ∈ B ψ → χ))

Distinct variable group(s):   x,y,φ   χ,x,y   y,A

Proof of Theorem r19.23advv

Step	Hyp	Ref	Expression
1		r19.23advv.1	. . . . . 6 ⊢ (φ → ((x ∈ A ∧ y ∈ B) → (ψ → χ)))
2	1	exp3a 292	. . . . 5 ⊢ (φ → (x ∈ A → (y ∈ B → (ψ → χ))))
3	2	imp 277	. . . 4 ⊢ ((φ ∧ x ∈ A) → (y ∈ B → (ψ → χ)))
4	3	r19.23adv 1286	. . 3 ⊢ ((φ ∧ x ∈ A) → (∃y ∈ B ψ → χ))
5	4	exp 291	. 2 ⊢ (φ → (x ∈ A → (∃y ∈ B ψ → χ)))
6	5	r19.23adv 1286	1 ⊢ (φ → (∃x ∈ A ∃y ∈ B ψ → χ))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  xpdom2 3345  unxpdomlem 3649  qaddclt 4642  qmulclt 4644  xpnnen 4927  infxpidmlem7 4939  shselt 5280  shmods 5363  sumdmdlem 5786

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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