Theorem cbvral2v 3367

Description: Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker cbvral2vw 3240 when possible. (Contributed by NM, 10-Aug-2004.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbvral2v.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
cbvral2v.2	⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
cbvral2v	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)

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

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

Proof of Theorem cbvral2v

Step	Hyp	Ref	Expression
1		cbvral2v.1	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
2	1	ralbidv 3177	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜒))
3	2	cbvralv 3363	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)
4		cbvral2v.2	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
5	4	cbvralv 3363	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑤 ∈ 𝐵 𝜓)
6	5	ralbii 3092	. 2 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
7	3, 6	bitri 275	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clel 2815  df-nfc 2891  df-ral 3061

This theorem is referenced by:  cbvral3v  3369