Theorem cbvral2 44546

Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3396. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Hypotheses

Ref	Expression
cbvral2.1	⊢ Ⅎ𝑧𝜑
cbvral2.2	⊢ Ⅎ𝑥𝜒
cbvral2.3	⊢ Ⅎ𝑤𝜒
cbvral2.4	⊢ Ⅎ𝑦𝜓
cbvral2.5	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
cbvral2.6	⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
cbvral2	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑧,𝐴   𝑥,𝑦,𝐵,𝑧   𝑦,𝑤,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)   𝜒(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑦,𝑤)

Proof of Theorem cbvral2

Step	Hyp	Ref	Expression
1		nfcv 2908	. . . 4 ⊢ Ⅎ𝑧𝐵
2		cbvral2.1	. . . 4 ⊢ Ⅎ𝑧𝜑
3	1, 2	nfral 3152	. . 3 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝐵 𝜑
4		nfcv 2908	. . . 4 ⊢ Ⅎ𝑥𝐵
5		cbvral2.2	. . . 4 ⊢ Ⅎ𝑥𝜒
6	4, 5	nfral 3152	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 𝜒
7		cbvral2.5	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
8	7	ralbidv 3122	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜒))
9	3, 6, 8	cbvralw 3371	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)
10		cbvral2.3	. . . 4 ⊢ Ⅎ𝑤𝜒
11		cbvral2.4	. . . 4 ⊢ Ⅎ𝑦𝜓
12		cbvral2.6	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
13	10, 11, 12	cbvralw 3371	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑤 ∈ 𝐵 𝜓)
14	13	ralbii 3092	. 2 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
15	9, 14	bitri 274	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 205  Ⅎwnf 1789  ∀wral 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-13 2373  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070

This theorem is referenced by: (None)