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Theorem sbequi 1764
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 1762 . . . 4 (∀𝑧 𝑧 = 𝑥 ∨ Ⅎ𝑧[𝑥 / 𝑧]𝜑)
2 nfr 1454 . . . . . 6 (Ⅎ𝑧[𝑥 / 𝑧]𝜑 → ([𝑥 / 𝑧]𝜑 → ∀𝑧[𝑥 / 𝑧]𝜑))
3 equvini 1685 . . . . . . 7 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
4 stdpc7 1697 . . . . . . . . 9 (𝑥 = 𝑧 → ([𝑥 / 𝑧]𝜑𝜑))
5 sbequ1 1695 . . . . . . . . 9 (𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
64, 5sylan9 401 . . . . . . . 8 ((𝑥 = 𝑧𝑧 = 𝑦) → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
76eximi 1534 . . . . . . 7 (∃𝑧(𝑥 = 𝑧𝑧 = 𝑦) → ∃𝑧([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
8 19.35-1 1558 . . . . . . 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 71 . . . . 5 (Ⅎ𝑧[𝑥 / 𝑧]𝜑 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → ∃𝑧[𝑦 / 𝑧]𝜑)))
1110orim2i 711 . . . 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 1762 . . . . 5 (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑧]𝜑)
14 19.9t 1576 . . . . . . 7 (Ⅎ𝑧[𝑦 / 𝑧]𝜑 → (∃𝑧[𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
1514biimpd 142 . . . . . 6 (Ⅎ𝑧[𝑦 / 𝑧]𝜑 → (∃𝑧[𝑦 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
1615orim2i 711 . . . . 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 711 . . . 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 1763 . 2 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))))
22 sbequ2 1696 . . . . . . 7 (𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑𝜑))
2322sps 1473 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑𝜑))
2423adantr 270 . . . . 5 ((∀𝑧 𝑧 = 𝑥𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑𝜑))
25 sbequ1 1695 . . . . . 6 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
26 drsb1 1724 . . . . . . . 8 (∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]𝜑))
2726biimpd 142 . . . . . . 7 (∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑧]𝜑))
2827alequcoms 1452 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑧]𝜑))
2925, 28sylan9r 402 . . . . 5 ((∀𝑧 𝑧 = 𝑥𝑥 = 𝑦) → (𝜑 → [𝑦 / 𝑧]𝜑))
3024, 29syld 44 . . . 4 ((∀𝑧 𝑧 = 𝑥𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
3130ex 113 . . 3 (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
32 drsb1 1724 . . . . . . . . 9 (∀𝑧 𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑥 / 𝑦]𝜑))
3332biimpd 142 . . . . . . . 8 (∀𝑧 𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑥 / 𝑦]𝜑))
34 stdpc7 1697 . . . . . . . 8 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
3533, 34sylan9 401 . . . . . . 7 ((∀𝑧 𝑧 = 𝑦𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑𝜑))
365sps 1473 . . . . . . . 8 (∀𝑧 𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
3736adantr 270 . . . . . . 7 ((∀𝑧 𝑧 = 𝑦𝑥 = 𝑦) → (𝜑 → [𝑦 / 𝑧]𝜑))
3835, 37syld 44 . . . . . 6 ((∀𝑧 𝑧 = 𝑦𝑥 = 𝑦) → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
3938ex 113 . . . . 5 (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
4039orim1i 710 . . . 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 pm1.2 706 . . . 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 669 . 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 102  wo 662  wal 1285  wnf 1392  wex 1424  [wsb 1689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690
This theorem is referenced by:  sbequ  1765
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