Theorem gencbvex2 2777

Description: Restatement of gencbvex 2776 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 296	. . . 4 ⊢ ((𝜒 ∧ 𝐴 = 𝑦) → 𝜃)
6	5	exlimiv 1591	. . 3 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝑦) → 𝜃)
7	4, 6	impbii 125	. 2 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))
8	1, 2, 3, 7	gencbvex 2776	1 ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by: (None)