Theorem cbvral2vw 3408

Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 3411 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 10-Aug-2004.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvral2vw.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
cbvral2vw.2	⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
cbvral2vw	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑤   𝑥,𝐴,𝑧   𝑥,𝑦,𝐵,𝑧   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 cbvral2vw

Step	Hyp	Ref	Expression
1		cbvral2vw.1	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
2	1	ralbidv 3162	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜒))
3	2	cbvralvw 3396	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)
4		cbvral2vw.2	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
5	4	cbvralvw 3396	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑤 ∈ 𝐵 𝜓)
6	5	ralbii 3133	. 2 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
7	3, 6	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∀wral 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clel 2870  df-ral 3111

This theorem is referenced by:  cbvral3vw  3410  fununi  6399  fiint  8779  nqereu  10340  mhmpropd  17954  efgred  18866  mplcoe5  20708  mdetunilem9  21225  fbun  22445  fbunfip  22474  caucfil  23887  pmltpc  24054  iscgrglt  26308  axcontlem10  26767  htth  28701  cdj3lem3b  30223  cdj3i  30224  dfmgc2  30704  isros  31537  rossros  31549  fipjust  40264  isotone1  40751  isotone2  40752  ntrclsiso  40770  ntrclskb  40772  ntrclsk3  40773  ntrclsk13  40774  limsuppnfd  42344  pimincfltioo  43353  incsmf  43376  decsmf  43400  mgmhmpropd  44405