Theorem cbvral2vw 3245

Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 3356 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 10-Aug-2004.) Avoid ax-13 2404. (Revised by GG, 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 3186	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜒))
3	2	cbvralvw 3241	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)
4		cbvral2vw.2	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
5	4	cbvralvw 3241	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑤 ∈ 𝐵 𝜓)
6	5	ralbii 3109	. 2 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
7	3, 6	bitri 277	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wral 3077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-clel 2838  df-ral 3078

This theorem is referenced by:  cbvral3vw  3247  cbvral6vw  3249  fununi  6597  fiint  9272  nqereu  10888  mgmhmpropd  18733  mhmpropd  18827  efgred  19789  mplcoe5  22094  mdetunilem9  22681  fbun  23901  fbunfip  23930  caucfil  25346  pmltpc  25513  negsprop  28129  iscgrglt  28684  axcontlem10  29175  htth  31122  cdj3lem3b  32644  cdj3i  32645  dfmgc2  33175  isros  34466  rossros  34478  fipjust  44142  isotone1  44625  isotone2  44626  ntrclsiso  44644  ntrclskb  44646  ntrclsk3  44647  ntrclsk13  44648  limsuppnfd  46277  pimincfltioo  47293  incsmf  47317  decsmf  47342  catprslem  49632  isthincd2lem1  50047  isthincd2lem2  50057