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Theorem cbvrald 13164
Description: Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
Hypotheses
Ref Expression
cbvrald.nf0 𝑥𝜑
cbvrald.nf1 𝑦𝜑
cbvrald.nf2 (𝜑 → Ⅎ𝑦𝜓)
cbvrald.nf3 (𝜑 → Ⅎ𝑥𝜒)
cbvrald.is (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbvrald (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐴 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem cbvrald
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvrald.nf0 . . . 4 𝑥𝜑
2 nfv 1509 . . . 4 𝑧𝜑
3 nfv 1509 . . . . . 6 𝑧 𝑥𝐴
43a1i 9 . . . . 5 (𝜑 → Ⅎ𝑧 𝑥𝐴)
5 nfv 1509 . . . . . 6 𝑧𝜓
65a1i 9 . . . . 5 (𝜑 → Ⅎ𝑧𝜓)
74, 6nfimd 1565 . . . 4 (𝜑 → Ⅎ𝑧(𝑥𝐴𝜓))
8 nfv 1509 . . . . . 6 𝑥 𝑧𝐴
98a1i 9 . . . . 5 (𝜑 → Ⅎ𝑥 𝑧𝐴)
10 nfs1v 1913 . . . . . 6 𝑥[𝑧 / 𝑥]𝜓
1110a1i 9 . . . . 5 (𝜑 → Ⅎ𝑥[𝑧 / 𝑥]𝜓)
129, 11nfimd 1565 . . . 4 (𝜑 → Ⅎ𝑥(𝑧𝐴 → [𝑧 / 𝑥]𝜓))
13 eleq1 2203 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1413adantl 275 . . . . . 6 ((𝜑𝑥 = 𝑧) → (𝑥𝐴𝑧𝐴))
15 sbequ12 1745 . . . . . . 7 (𝑥 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑥]𝜓))
1615adantl 275 . . . . . 6 ((𝜑𝑥 = 𝑧) → (𝜓 ↔ [𝑧 / 𝑥]𝜓))
1714, 16imbi12d 233 . . . . 5 ((𝜑𝑥 = 𝑧) → ((𝑥𝐴𝜓) ↔ (𝑧𝐴 → [𝑧 / 𝑥]𝜓)))
1817ex 114 . . . 4 (𝜑 → (𝑥 = 𝑧 → ((𝑥𝐴𝜓) ↔ (𝑧𝐴 → [𝑧 / 𝑥]𝜓))))
191, 2, 7, 12, 18cbv2 1726 . . 3 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜓)))
20 cbvrald.nf1 . . . 4 𝑦𝜑
21 nfv 1509 . . . . . 6 𝑦 𝑧𝐴
2221a1i 9 . . . . 5 (𝜑 → Ⅎ𝑦 𝑧𝐴)
23 cbvrald.nf2 . . . . . 6 (𝜑 → Ⅎ𝑦𝜓)
241, 23nfsbd 1951 . . . . 5 (𝜑 → Ⅎ𝑦[𝑧 / 𝑥]𝜓)
2522, 24nfimd 1565 . . . 4 (𝜑 → Ⅎ𝑦(𝑧𝐴 → [𝑧 / 𝑥]𝜓))
26 nfv 1509 . . . . . 6 𝑧 𝑦𝐴
2726a1i 9 . . . . 5 (𝜑 → Ⅎ𝑧 𝑦𝐴)
28 nfv 1509 . . . . . 6 𝑧𝜒
2928a1i 9 . . . . 5 (𝜑 → Ⅎ𝑧𝜒)
3027, 29nfimd 1565 . . . 4 (𝜑 → Ⅎ𝑧(𝑦𝐴𝜒))
31 eleq1 2203 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
3231adantl 275 . . . . . 6 ((𝜑𝑧 = 𝑦) → (𝑧𝐴𝑦𝐴))
33 sbequ 1813 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
34 cbvrald.nf3 . . . . . . . 8 (𝜑 → Ⅎ𝑥𝜒)
35 cbvrald.is . . . . . . . 8 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
361, 34, 35sbied 1762 . . . . . . 7 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
3733, 36sylan9bbr 459 . . . . . 6 ((𝜑𝑧 = 𝑦) → ([𝑧 / 𝑥]𝜓𝜒))
3832, 37imbi12d 233 . . . . 5 ((𝜑𝑧 = 𝑦) → ((𝑧𝐴 → [𝑧 / 𝑥]𝜓) ↔ (𝑦𝐴𝜒)))
3938ex 114 . . . 4 (𝜑 → (𝑧 = 𝑦 → ((𝑧𝐴 → [𝑧 / 𝑥]𝜓) ↔ (𝑦𝐴𝜒))))
402, 20, 25, 30, 39cbv2 1726 . . 3 (𝜑 → (∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜓) ↔ ∀𝑦(𝑦𝐴𝜒)))
4119, 40bitrd 187 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑦(𝑦𝐴𝜒)))
42 df-ral 2422 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
43 df-ral 2422 . 2 (∀𝑦𝐴 𝜒 ↔ ∀𝑦(𝑦𝐴𝜒))
4441, 42, 433bitr4g 222 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1330  wnf 1437  wcel 1481  [wsb 1736  wral 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-ral 2422
This theorem is referenced by:  setindft  13332
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