Theorem cbvral3v 3390

Description: Change bound variables of triple restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvral3vw 3387 when possible. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbvral3v.1	⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))
cbvral3v.2	⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃))
cbvral3v.3	⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓))

Assertion

Ref	Expression
cbvral3v	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)

Distinct variable groups:   𝜑,𝑤   𝜓,𝑧   𝜒,𝑥   𝜒,𝑣   𝜃,𝑦   𝜃,𝑢   𝑥,𝐴   𝑤,𝐴   𝑥,𝑦,𝐵   𝑦,𝑤,𝐵   𝑣,𝐵   𝑥,𝑧,𝐶,𝑦   𝑧,𝑤,𝐶   𝑧,𝑣,𝐶   𝑢,𝐶

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑣,𝑢)   𝜓(𝑥,𝑦,𝑤,𝑣,𝑢)   𝜒(𝑦,𝑧,𝑤,𝑢)   𝜃(𝑥,𝑧,𝑤,𝑣)   𝐴(𝑦,𝑧,𝑣,𝑢)   𝐵(𝑧,𝑢)

Proof of Theorem cbvral3v

Step	Hyp	Ref	Expression
1		cbvral3v.1	. . . 4 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))
2	1	2ralbidv 3122	. . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒))
3	2	cbvralv 3377	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒)
4		cbvral3v.2	. . . 4 ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃))
5		cbvral3v.3	. . . 4 ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓))
6	4, 5	cbvral2v 3388	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)
7	6	ralbii 3090	. 2 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)
8	3, 7	bitri 274	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clel 2817  df-nfc 2888  df-ral 3068

This theorem is referenced by: (None)