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Theorem sbal1yz 1893
Description: Lemma for proving sbal1 1894. Same as sbal1 1894 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 1713 . . . . . 6 (∀𝑥 𝑥 = 𝑧 ∨ Ⅎ𝑥 𝑦 = 𝑧)
2 equcom 1609 . . . . . . . . 9 (𝑦 = 𝑧𝑧 = 𝑦)
32nfbii 1378 . . . . . . . 8 (Ⅎ𝑥 𝑦 = 𝑧 ↔ Ⅎ𝑥 𝑧 = 𝑦)
4 19.21t 1490 . . . . . . . 8 (Ⅎ𝑥 𝑧 = 𝑦 → (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
53, 4sylbi 118 . . . . . . 7 (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
65orim2i 688 . . . . . 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 652 . . . 4 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
98albidv 1721 . . 3 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑦𝑥(𝑧 = 𝑦𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑)))
10 alcom 1383 . . . 4 (∀𝑦𝑥(𝑧 = 𝑦𝜑) ↔ ∀𝑥𝑦(𝑧 = 𝑦𝜑))
11 sb6 1782 . . . . . 6 ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧𝜑))
122imbi1i 231 . . . . . . 7 ((𝑦 = 𝑧𝜑) ↔ (𝑧 = 𝑦𝜑))
1312albii 1375 . . . . . 6 (∀𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦(𝑧 = 𝑦𝜑))
1411, 13bitri 177 . . . . 5 ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑧 = 𝑦𝜑))
1514albii 1375 . . . 4 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥𝑦(𝑧 = 𝑦𝜑))
1610, 15bitr4i 180 . . 3 (∀𝑦𝑥(𝑧 = 𝑦𝜑) ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
17 sb6 1782 . . . 4 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑))
182imbi1i 231 . . . . 5 ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑))
1918albii 1375 . . . 4 (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑))
2017, 19bitr2i 178 . . 3 (∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑) ↔ [𝑧 / 𝑦]∀𝑥𝜑)
219, 16, 203bitr3g 215 . 2 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
2221bicomd 133 1 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  wal 1257  wnf 1365  [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-in2 555  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:  sbal1  1894
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