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Theorem cbvral2v 2598
Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
Hypotheses
Ref Expression
cbvral2v.1 (𝑥 = 𝑧 → (𝜑𝜒))
cbvral2v.2 (𝑦 = 𝑤 → (𝜒𝜓))
Assertion
Ref Expression
cbvral2v (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑧,𝐴   𝑥,𝑦,𝐵   𝑦,𝑧,𝐵   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 cbvral2v
StepHypRef Expression
1 cbvral2v.1 . . . 4 (𝑥 = 𝑧 → (𝜑𝜒))
21ralbidv 2380 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 𝜒))
32cbvralv 2590 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑦𝐵 𝜒)
4 cbvral2v.2 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
54cbvralv 2590 . . 3 (∀𝑦𝐵 𝜒 ↔ ∀𝑤𝐵 𝜓)
65ralbii 2384 . 2 (∀𝑧𝐴𝑦𝐵 𝜒 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
73, 6bitri 182 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wral 2359
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364
This theorem is referenced by:  cbvral3v  2600  fununi  5082  fiintim  6637  isoti  6700  cauappcvgprlemlim  7218  caucvgprlemnkj  7223  caucvgprlemcl  7233  caucvgprprlemcbv  7244  axcaucvglemcau  7431  iseqdistr  9941  seq3distr  9942
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