Theorem cla4e2gv 1398

Description: Existential specialization with 2 quantifiers, using implicit substitution.

Hypothesis

Ref	Expression
cla4e2gv.1	⊢ ((x = A ∧ y = B) → (φ ↔ ψ))

Assertion

Ref	Expression
cla4e2gv	⊢ ((A ∈ C ∧ B ∈ D) → (ψ → ∃x∃yφ))

Distinct variable group(s):   x,y,A   x,B,y   ψ,x,y

Proof of Theorem cla4e2gv

Step	Hyp	Ref	Expression
1		cla4e2gv.1	. . . . 5 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
2	1	biimprcd 138	. . . 4 ⊢ (ψ → ((x = A ∧ y = B) → φ))
3	2	19.22dvv 949	. . 3 ⊢ (ψ → (∃x∃y(x = A ∧ y = B) → ∃x∃yφ))
4		elex 1356	. . . . 5 ⊢ (A ∈ C → ∃x x = A)
5		elex 1356	. . . . 5 ⊢ (B ∈ D → ∃y y = B)
6	4, 5	anim12i 268	. . . 4 ⊢ ((A ∈ C ∧ B ∈ D) → (∃x x = A ∧ ∃y y = B))
7		eeanv 980	. . . 4 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
8	6, 7	sylibr 175	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) → ∃x∃y(x = A ∧ y = B))
9	3, 8	syl5 22	. 2 ⊢ (ψ → ((A ∈ C ∧ B ∈ D) → ∃x∃yφ))
10	9	com12 13	1 ⊢ ((A ∈ C ∧ B ∈ D) → (ψ → ∃x∃yφ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  cla42gv 1399  cla4e2v 1406  th3q 3253  genpprecl 3898

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-7 676  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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