Theorem gencbvex2 2784

Description: Restatement of gencbvex 2783 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 298	. . . 4 ⊢ ((𝜒 ∧ 𝐴 = 𝑦) → 𝜃)
6	5	exlimiv 1598	. . 3 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝑦) → 𝜃)
7	4, 6	impbii 126	. 2 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))
8	1, 2, 3, 7	gencbvex 2783	1 ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739

This theorem is referenced by: (None)