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Theorem sbal1yz 1877
 Description: Lemma for proving sbal1 1878. Same as sbal1 1878 but with an additional distinct variable constraint on 𝑦 and 𝑧. (Contributed by Jim Kingdon, 23-Feb-2018.)
Assertion
Ref Expression
sbal1yz (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbal1yz
StepHypRef Expression
1 dveeq2or 1697 . . . . . 6 (∀𝑥 𝑥 = 𝑧 ∨ Ⅎ𝑥 𝑦 = 𝑧)
2 equcom 1593 . . . . . . . . 9 (𝑦 = 𝑧𝑧 = 𝑦)
32nfbii 1362 . . . . . . . 8 (Ⅎ𝑥 𝑦 = 𝑧 ↔ Ⅎ𝑥 𝑧 = 𝑦)
4 19.21t 1474 . . . . . . . 8 (Ⅎ𝑥 𝑧 = 𝑦 → (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
53, 4sylbi 114 . . . . . . 7 (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
65orim2i 678 . . . . . 6 ((∀𝑥 𝑥 = 𝑧 ∨ Ⅎ𝑥 𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑))))
71, 6ax-mp 7 . . . . 5 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
87ori 642 . . . 4 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
98albidv 1705 . . 3 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑦𝑥(𝑧 = 𝑦𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑)))
10 alcom 1367 . . . 4 (∀𝑦𝑥(𝑧 = 𝑦𝜑) ↔ ∀𝑥𝑦(𝑧 = 𝑦𝜑))
11 sb6 1766 . . . . . 6 ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧𝜑))
122imbi1i 227 . . . . . . 7 ((𝑦 = 𝑧𝜑) ↔ (𝑧 = 𝑦𝜑))
1312albii 1359 . . . . . 6 (∀𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦(𝑧 = 𝑦𝜑))
1411, 13bitri 173 . . . . 5 ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑧 = 𝑦𝜑))
1514albii 1359 . . . 4 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥𝑦(𝑧 = 𝑦𝜑))
1610, 15bitr4i 176 . . 3 (∀𝑦𝑥(𝑧 = 𝑦𝜑) ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
17 sb6 1766 . . . 4 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑))
182imbi1i 227 . . . . 5 ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑))
1918albii 1359 . . . 4 (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑))
2017, 19bitr2i 174 . . 3 (∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑) ↔ [𝑧 / 𝑦]∀𝑥𝜑)
219, 16, 203bitr3g 211 . 2 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
2221bicomd 129 1 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 629  ∀wal 1241  Ⅎwnf 1349  [wsb 1645 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-in2 545  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-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  sbal1  1878
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