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Theorem cbvralf 2527
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvralf.1 𝑥𝐴
cbvralf.2 𝑦𝐴
cbvralf.3 𝑦𝜑
cbvralf.4 𝑥𝜓
cbvralf.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvralf (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Proof of Theorem cbvralf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1421 . . . 4 𝑧(𝑥𝐴𝜑)
2 cbvralf.1 . . . . . 6 𝑥𝐴
32nfcri 2172 . . . . 5 𝑥 𝑧𝐴
4 nfs1v 1815 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfim 1464 . . . 4 𝑥(𝑧𝐴 → [𝑧 / 𝑥]𝜑)
6 eleq1 2100 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7 sbequ12 1654 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
86, 7imbi12d 223 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 → [𝑧 / 𝑥]𝜑)))
91, 5, 8cbval 1637 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜑))
10 cbvralf.2 . . . . . 6 𝑦𝐴
1110nfcri 2172 . . . . 5 𝑦 𝑧𝐴
12 cbvralf.3 . . . . . 6 𝑦𝜑
1312nfsb 1822 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1411, 13nfim 1464 . . . 4 𝑦(𝑧𝐴 → [𝑧 / 𝑥]𝜑)
15 nfv 1421 . . . 4 𝑧(𝑦𝐴𝜓)
16 eleq1 2100 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
17 sbequ 1721 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18 cbvralf.4 . . . . . . 7 𝑥𝜓
19 cbvralf.5 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
2018, 19sbie 1674 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
2117, 20syl6bb 185 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
2216, 21imbi12d 223 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 → [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2314, 15, 22cbval 1637 . . 3 (∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
249, 23bitri 173 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
25 df-ral 2311 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
26 df-ral 2311 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2724, 25, 263bitr4i 201 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349   ∈ wcel 1393  [wsb 1645  Ⅎwnfc 2165  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311 This theorem is referenced by:  cbvral  2529  ffnfvf  5324
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