Theorem gencbvex2 2903

Description: Restatement of gencbvex 2902 with weaker hypotheses. (Contributed by Jeffrey Hankins, 6-Dec-2006.)

Hypotheses

Ref	Expression
gencbvex2.1	⊢ A ∈ V
gencbvex2.2	⊢ (A = y → (φ ↔ ψ))
gencbvex2.3	⊢ (A = y → (χ ↔ θ))
gencbvex2.4	⊢ (θ → ∃x(χ ∧ A = y))

Assertion

Ref	Expression
gencbvex2	⊢ (∃x(χ ∧ φ) ↔ ∃y(θ ∧ ψ))

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

Allowed substitution hints:   φ(x)   ψ(y)   χ(x)   θ(y)   A(x)

Proof of Theorem gencbvex2

Step	Hyp	Ref	Expression
1		gencbvex2.1	. 2 ⊢ A ∈ V
2		gencbvex2.2	. 2 ⊢ (A = y → (φ ↔ ψ))
3		gencbvex2.3	. 2 ⊢ (A = y → (χ ↔ θ))
4		gencbvex2.4	. . 3 ⊢ (θ → ∃x(χ ∧ A = y))
5	3	biimpac 472	. . . 4 ⊢ ((χ ∧ A = y) → θ)
6	5	exlimiv 1634	. . 3 ⊢ (∃x(χ ∧ A = y) → θ)
7	4, 6	impbii 180	. 2 ⊢ (θ ↔ ∃x(χ ∧ A = y))
8	1, 2, 3, 7	gencbvex 2902	1 ⊢ (∃x(χ ∧ φ) ↔ ∃y(θ ∧ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860

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: (None)