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Theorem cbvralfwOLD 3367
Description: Obsolete version of cbvralfw 3366 as of 23-May-2024. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvralfwOLD.1 𝑥𝐴
cbvralfwOLD.2 𝑦𝐴
cbvralfwOLD.3 𝑦𝜑
cbvralfwOLD.4 𝑥𝜓
cbvralfwOLD.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvralfwOLD (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem cbvralfwOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1920 . . . 4 𝑧(𝑥𝐴𝜑)
2 cbvralfwOLD.1 . . . . . 6 𝑥𝐴
32nfcri 2895 . . . . 5 𝑥 𝑧𝐴
4 nfs1v 2156 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfim 1902 . . . 4 𝑥(𝑧𝐴 → [𝑧 / 𝑥]𝜑)
6 eleq1w 2822 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7 sbequ12 2247 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
86, 7imbi12d 344 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 → [𝑧 / 𝑥]𝜑)))
91, 5, 8cbvalv1 2341 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜑))
10 cbvralfwOLD.2 . . . . . 6 𝑦𝐴
1110nfcri 2895 . . . . 5 𝑦 𝑧𝐴
12 cbvralfwOLD.3 . . . . . 6 𝑦𝜑
1312nfsbv 2327 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1411, 13nfim 1902 . . . 4 𝑦(𝑧𝐴 → [𝑧 / 𝑥]𝜑)
15 nfv 1920 . . . 4 𝑧(𝑦𝐴𝜓)
16 eleq1w 2822 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
17 sbequ 2089 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18 cbvralfwOLD.4 . . . . . . 7 𝑥𝜓
19 cbvralfwOLD.5 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
2018, 19sbiev 2312 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
2117, 20bitrdi 286 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
2216, 21imbi12d 344 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 → [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2314, 15, 22cbvalv1 2341 . . 3 (∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
249, 23bitri 274 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
25 df-ral 3070 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
26 df-ral 3070 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2724, 25, 263bitr4i 302 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wnf 1789  [wsb 2070  wcel 2109  wnfc 2888  wral 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-10 2140  ax-11 2157  ax-12 2174
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1786  df-nf 1790  df-sb 2071  df-clel 2817  df-nfc 2890  df-ral 3070
This theorem is referenced by: (None)
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