Theorem cbvral3v 2587

Description: Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)

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   𝑦,𝑢,𝜃   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 2390	. . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒))
3	2	cbvralv 2577	. 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 2585	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)
7	6	ralbii 2372	. 2 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)
8	3, 7	bitri 182	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ↔ wb 103  ∀wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353

This theorem is referenced by: (None)