Theorem cbvraldva2 3455

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 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
2		cbvraldva2.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
3	1, 2	eleq12d 2905	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4		cbvraldva2.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
5	3, 4	imbi12d 347	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑦 ∈ 𝐵 → 𝜒)))
6	5	expcom 416	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑦 ∈ 𝐵 → 𝜒))))
7	6	pm5.74d 275	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → (𝑦 ∈ 𝐵 → 𝜒))))
8	7	cbvalvw 2037	. . . 4 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ ∀𝑦(𝜑 → (𝑦 ∈ 𝐵 → 𝜒)))
9		19.21v 1934	. . . 4 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
10		19.21v 1934	. . . 4 ⊢ (∀𝑦(𝜑 → (𝑦 ∈ 𝐵 → 𝜒)) ↔ (𝜑 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜒)))
11	8, 9, 10	3bitr3i 303	. . 3 ⊢ ((𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜒)))
12	11	pm5.74ri 274	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜒)))
13		df-ral 3141	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
14		df-ral 3141	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜒))
15	12, 13, 14	3bitr4g 316	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1529   = wceq 1531   ∈ wcel 2108  ∀wral 3136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-cleq 2812  df-clel 2891  df-ral 3141

This theorem is referenced by:  cbvraldva  3458  tfrlem3a  8005  mreexexlemd  16907  ismnu  40587