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Theorem sbequi 1736
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)
Assertion
Ref Expression
sbequi (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Proof of Theorem sbequi
StepHypRef Expression
1 nfsb2or 1734 . . . 4 (∀𝑧 𝑧 = 𝑥 ∨ Ⅎ𝑧[𝑥 / 𝑧]𝜑)
2 nfr 1427 . . . . . 6 (Ⅎ𝑧[𝑥 / 𝑧]𝜑 → ([𝑥 / 𝑧]𝜑 → ∀𝑧[𝑥 / 𝑧]𝜑))
3 equvini 1657 . . . . . . 7 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
4 stdpc7 1669 . . . . . . . . 9 (𝑥 = 𝑧 → ([𝑥 / 𝑧]𝜑𝜑))
5 sbequ1 1667 . . . . . . . . 9 (𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
64, 5sylan9 395 . . . . . . . 8 ((𝑥 = 𝑧𝑧 = 𝑦) → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
76eximi 1507 . . . . . . 7 (∃𝑧(𝑥 = 𝑧𝑧 = 𝑦) → ∃𝑧([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
8 19.35-1 1531 . . . . . . 7 (∃𝑧([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑) → (∀𝑧[𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑))
93, 7, 83syl 17 . . . . . 6 (𝑥 = 𝑦 → (∀𝑧[𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑))
102, 9syl9 70 . . . . 5 (Ⅎ𝑧[𝑥 / 𝑧]𝜑 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑)))
1110orim2i 688 . . . 4 ((∀𝑧 𝑧 = 𝑥 ∨ Ⅎ𝑧[𝑥 / 𝑧]𝜑) → (∀𝑧 𝑧 = 𝑥 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑))))
121, 11ax-mp 7 . . 3 (∀𝑧 𝑧 = 𝑥 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑)))
13 nfsb2or 1734 . . . . 5 (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑧]𝜑)
14 19.9t 1549 . . . . . . 7 (Ⅎ𝑧[𝑦 / 𝑧]𝜑 → (∃𝑧[𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
1514biimpd 136 . . . . . 6 (Ⅎ𝑧[𝑦 / 𝑧]𝜑 → (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
1615orim2i 688 . . . . 5 ((∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑧]𝜑) → (∀𝑧 𝑧 = 𝑦 ∨ (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
1713, 16ax-mp 7 . . . 4 (∀𝑧 𝑧 = 𝑦 ∨ (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
18 ax-1 5 . . . . 5 ((∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑) → (𝑥 = 𝑦 → (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
1918orim2i 688 . . . 4 ((∀𝑧 𝑧 = 𝑦 ∨ (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))))
2017, 19ax-mp 7 . . 3 (∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
2112, 20sbequilem 1735 . 2 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))))
22 sbequ2 1668 . . . . . . 7 (𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑𝜑))
2322sps 1446 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑𝜑))
2423adantr 265 . . . . 5 ((∀𝑧 𝑧 = 𝑥𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑𝜑))
25 sbequ1 1667 . . . . . 6 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
26 drsb1 1696 . . . . . . . 8 (∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]𝜑))
2726biimpd 136 . . . . . . 7 (∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑧]𝜑))
2827alequcoms 1425 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑧]𝜑))
2925, 28sylan9r 396 . . . . 5 ((∀𝑧 𝑧 = 𝑥𝑥 = 𝑦) → (𝜑 → [𝑦 / 𝑧]𝜑))
3024, 29syld 44 . . . 4 ((∀𝑧 𝑧 = 𝑥𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
3130ex 112 . . 3 (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
32 drsb1 1696 . . . . . . . . 9 (∀𝑧 𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑥 / 𝑦]𝜑))
3332biimpd 136 . . . . . . . 8 (∀𝑧 𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑥 / 𝑦]𝜑))
34 stdpc7 1669 . . . . . . . 8 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
3533, 34sylan9 395 . . . . . . 7 ((∀𝑧 𝑧 = 𝑦𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑𝜑))
365sps 1446 . . . . . . . 8 (∀𝑧 𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
3736adantr 265 . . . . . . 7 ((∀𝑧 𝑧 = 𝑦𝑥 = 𝑦) → (𝜑 → [𝑦 / 𝑧]𝜑))
3835, 37syld 44 . . . . . 6 ((∀𝑧 𝑧 = 𝑦𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
3938ex 112 . . . . 5 (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
4039orim1i 687 . . . 4 ((∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))) → ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))))
41 pm1.2 683 . . . 4 (((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))) → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
4240, 41syl 14 . . 3 ((∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))) → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
4331, 42jaoi 646 . 2 ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))) → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
4421, 43ax-mp 7 1 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wo 639  wal 1257  wnf 1365  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  sbequ  1737
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