Theorem cbvrexdva2OLD 3405

Description: Obsolete version of cbvrexdva 3407 as of 12-Aug-2023. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbvraldva2.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
cbvraldva2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvrexdva2OLD	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))

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

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrexdva2OLD

Step	Hyp	Ref	Expression
1		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
2		cbvraldva2.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
3	1, 2	eleq12d 2884	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4		cbvraldva2.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
5	3, 4	anbi12d 633	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))
6	5	notbid 321	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (¬ (𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ¬ (𝑦 ∈ 𝐵 ∧ 𝜒)))
7	6	expcom 417	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 → (¬ (𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ¬ (𝑦 ∈ 𝐵 ∧ 𝜒))))
8	7	pm5.74d 276	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜑 → ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 → ¬ (𝑦 ∈ 𝐵 ∧ 𝜒))))
9	8	cbvalvw 2043	. . . . . 6 ⊢ (∀𝑥(𝜑 → ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ ∀𝑦(𝜑 → ¬ (𝑦 ∈ 𝐵 ∧ 𝜒)))
10		19.21v 1940	. . . . . 6 ⊢ (∀𝑥(𝜑 → ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 → ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)))
11		19.21v 1940	. . . . . 6 ⊢ (∀𝑦(𝜑 → ¬ (𝑦 ∈ 𝐵 ∧ 𝜒)) ↔ (𝜑 → ∀𝑦 ¬ (𝑦 ∈ 𝐵 ∧ 𝜒)))
12	9, 10, 11	3bitr3i 304	. . . . 5 ⊢ ((𝜑 → ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 → ∀𝑦 ¬ (𝑦 ∈ 𝐵 ∧ 𝜒)))
13	12	pm5.74ri 275	. . . 4 ⊢ (𝜑 → (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∀𝑦 ¬ (𝑦 ∈ 𝐵 ∧ 𝜒)))
14		alnex 1783	. . . 4 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
15		alnex 1783	. . . 4 ⊢ (∀𝑦 ¬ (𝑦 ∈ 𝐵 ∧ 𝜒) ↔ ¬ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒))
16	13, 14, 15	3bitr3g 316	. . 3 ⊢ (𝜑 → (¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ¬ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒)))
17	16	con4bid 320	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒)))
18		df-rex 3112	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
19		df-rex 3112	. 2 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒))
20	17, 18, 19	3bitr4g 317	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-rex 3112

This theorem is referenced by: (None)