Theorem cbvral2v 3358

Description: Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker cbvral2vw 3232 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 3171	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜒))
3	2	cbvralv 3354	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)
4		cbvral2v.2	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
5	4	cbvralv 3354	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑤 ∈ 𝐵 𝜓)
6	5	ralbii 3087	. 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 205  ∀wral 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clel 2804  df-nfc 2879  df-ral 3056

This theorem is referenced by:  cbvral3v  3360