Theorem gencbvex2 2646

Description: Restatement of gencbvex 2645 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 292	. . . 4 ⊢ ((𝜒 ∧ 𝐴 = 𝑦) → 𝜃)
6	5	exlimiv 1529	. . 3 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝑦) → 𝜃)
7	4, 6	impbii 124	. 2 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))
8	1, 2, 3, 7	gencbvex 2645	1 ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  ∃wex 1421   ∈ wcel 1433  Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by: (None)