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Theorem cbvralfw 3299
Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 3354 with a disjoint variable condition, which does not require ax-10 2135, ax-13 2369. For a version not dependent on ax-11 2152 and ax-12, see cbvralvw 3232. (Contributed by NM, 7-Mar-2004.) Avoid ax-10 2135, ax-13 2369. (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 2888 . . . 4 𝑦 𝑥𝐴
3 cbvralfw.3 . . . 4 𝑦𝜑
42, 3nfim 1897 . . 3 𝑦(𝑥𝐴𝜑)
5 cbvralfw.1 . . . . 5 𝑥𝐴
65nfcri 2888 . . . 4 𝑥 𝑦𝐴
7 cbvralfw.4 . . . 4 𝑥𝜓
86, 7nfim 1897 . . 3 𝑥(𝑦𝐴𝜓)
9 eleq1w 2814 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
10 cbvralfw.5 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
119, 10imbi12d 343 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
124, 8, 11cbvalv1 2335 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
13 df-ral 3060 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
14 df-ral 3060 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
1512, 13, 143bitr4i 302 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537  wnf 1783  wcel 2104  wnfc 2881  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-11 2152  ax-12 2169
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-nf 1784  df-clel 2808  df-nfc 2883  df-ral 3060
This theorem is referenced by:  cbvrexfw  3300  cbvralw  3301  reusv2lem4  5398  reusv2  5400  ffnfvf  7120  nnwof  12902  nnindf  32292  scottexf  37339  scott0f  37340  evth2f  44001  evthf  44013  fmptff  44272  supxrleubrnmptf  44459  stoweidlem14  45028  stoweidlem28  45042  stoweidlem59  45073
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