Theorem spc2egv 2942

Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Hypothesis

Ref	Expression
spc2egv.1	⊢ ((x = A ∧ y = B) → (φ ↔ ψ))

Assertion

Ref	Expression
spc2egv	⊢ ((A ∈ V ∧ B ∈ W) → (ψ → ∃x∃yφ))

Distinct variable groups:   x,y,A   x,B,y   ψ,x,y

Allowed substitution hints:   φ(x,y)   V(x,y)   W(x,y)

Proof of Theorem spc2egv

Step	Hyp	Ref	Expression
1		elisset 2870	. . . 4 ⊢ (A ∈ V → ∃x x = A)
2		elisset 2870	. . . 4 ⊢ (B ∈ W → ∃y y = B)
3	1, 2	anim12i 549	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → (∃x x = A ∧ ∃y y = B))
4		eeanv 1913	. . 3 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
5	3, 4	sylibr 203	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → ∃x∃y(x = A ∧ y = B))
6		spc2egv.1	. . . 4 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
7	6	biimprcd 216	. . 3 ⊢ (ψ → ((x = A ∧ y = B) → φ))
8	7	2eximdv 1624	. 2 ⊢ (ψ → (∃x∃y(x = A ∧ y = B) → ∃x∃yφ))
9	5, 8	syl5com 26	1 ⊢ ((A ∈ V ∧ B ∈ W) → (ψ → ∃x∃yφ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  spc2gv  2943  spc2ev  2948