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Theorem cbvralfw 3286
Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 3332 with a disjoint variable condition, which does not require ax-10 2138, ax-13 2371. For a version not dependent on ax-11 2155 and ax-12, see cbvralvw 3224. (Contributed by NM, 7-Mar-2004.) Avoid ax-10 2138, ax-13 2371. (Revised by Gino Giotto, 23-May-2024.)
Hypotheses
Ref Expression
cbvralfw.1 𝑥𝐴
cbvralfw.2 𝑦𝐴
cbvralfw.3 𝑦𝜑
cbvralfw.4 𝑥𝜓
cbvralfw.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvralfw (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem cbvralfw
StepHypRef Expression
1 cbvralfw.2 . . . . 5 𝑦𝐴
21nfcri 2891 . . . 4 𝑦 𝑥𝐴
3 cbvralfw.3 . . . 4 𝑦𝜑
42, 3nfim 1900 . . 3 𝑦(𝑥𝐴𝜑)
5 cbvralfw.1 . . . . 5 𝑥𝐴
65nfcri 2891 . . . 4 𝑥 𝑦𝐴
7 cbvralfw.4 . . . 4 𝑥𝜓
86, 7nfim 1900 . . 3 𝑥(𝑦𝐴𝜓)
9 eleq1w 2817 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
10 cbvralfw.5 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
119, 10imbi12d 345 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
124, 8, 11cbvalv1 2338 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
13 df-ral 3062 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
14 df-ral 3062 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
1512, 13, 143bitr4i 303 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  wnf 1786  wcel 2107  wnfc 2884  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-11 2155  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787  df-clel 2811  df-nfc 2886  df-ral 3062
This theorem is referenced by:  cbvrexfw  3287  cbvralw  3288  reusv2lem4  5357  reusv2  5359  ffnfvf  7068  nnwof  12844  nnindf  31764  scottexf  36673  scott0f  36674  evth2f  43308  evthf  43320  fmptff  43585  supxrleubrnmptf  43772  stoweidlem14  44341  stoweidlem28  44355  stoweidlem59  44386
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