Theorem cbvraldva2 3403

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

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

Assertion

Ref	Expression
cbvraldva2	⊢ (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 cbvraldva2

Step	Hyp	Ref	Expression
1		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
2		cbvraldva2.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
3	1, 2	eleq12d 2884	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4		cbvraldva2.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
5	3, 4	imbi12d 348	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑦 ∈ 𝐵 → 𝜒)))
6	5	expcom 417	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑦 ∈ 𝐵 → 𝜒))))
7	6	pm5.74d 276	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → (𝑦 ∈ 𝐵 → 𝜒))))
8	7	cbvalvw 2043	. . . 4 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ ∀𝑦(𝜑 → (𝑦 ∈ 𝐵 → 𝜒)))
9		19.21v 1940	. . . 4 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
10		19.21v 1940	. . . 4 ⊢ (∀𝑦(𝜑 → (𝑦 ∈ 𝐵 → 𝜒)) ↔ (𝜑 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜒)))
11	8, 9, 10	3bitr3i 304	. . 3 ⊢ ((𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜒)))
12	11	pm5.74ri 275	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜒)))
13		df-ral 3111	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
14		df-ral 3111	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜒))
15	12, 13, 14	3bitr4g 317	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3106

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-ral 3111

This theorem is referenced by:  cbvraldva  3406  tfrlem3a  7996  mreexexlemd  16907  ismnu  40969