Theorem gencbvex2 3489

Description: Restatement of gencbvex 3488 with weaker hypotheses. (Contributed by Jeff 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 479	. . . 4 ⊢ ((𝜒 ∧ 𝐴 = 𝑦) → 𝜃)
6	5	exlimiv 1933	. . 3 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝑦) → 𝜃)
7	4, 6	impbii 208	. 2 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))
8	1, 2, 3, 7	gencbvex 3488	1 ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-clel 2816

This theorem is referenced by: (None)