Theorem cbvral2vw 3244

Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 3353 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by NM, 10-Aug-2004.) Avoid ax-13 2401. (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 3185	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜒))
3	2	cbvralvw 3240	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)
4		cbvral2vw.2	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
5	4	cbvralvw 3240	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑤 ∈ 𝐵 𝜓)
6	5	ralbii 3108	. 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2835  df-ral 3077

This theorem is used by:  cbvral3vw  3246  cbvral6vw  3248  fununi  6609  fiint  9299  nqereu  10941  mgmhmpropd  18803  mhmpropd  18903  efgred  19878  mplcoe5  22259  mdetunilem9  22845  fbun  24069  fbunfip  24098  caucfil  25514  pmltpc  25681  negsprop  28303  iscgrglt  28859  axcontlem10  29433  htth  31402  cdj3lem3b  32924  cdj3i  32925  dfmgc2  33439  isros  34682  rossros  34694  nadddilem2  36804  nadddilem4  36806  fipjust  44408  isotone1  44891  isotone2  44892  ntrclsiso  44910  ntrclskb  44912  ntrclsk3  44913  ntrclsk13  44914  limsuppnfd  46533  pimincfltioo  47549  incsmf  47573  decsmf  47598  catprslem  49939  isthincd2lem1  50354  isthincd2lem2  50364