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Theorem sbal1 1878
Description: A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.)
Assertion
Ref Expression
sbal1 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbal1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sbal 1876 . . . 4 ([𝑤 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑤 / 𝑦]𝜑)
21sbbii 1648 . . 3 ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑤]∀𝑥[𝑤 / 𝑦]𝜑)
3 sbal1yz 1877 . . 3 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑤]∀𝑥[𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑))
42, 3syl5bb 181 . 2 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑))
5 ax-17 1419 . . 3 (∀𝑥𝜑 → ∀𝑤𝑥𝜑)
65sbco2v 1821 . 2 ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑)
7 ax-17 1419 . . . 4 (𝜑 → ∀𝑤𝜑)
87sbco2v 1821 . . 3 ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑)
98albii 1359 . 2 (∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
104, 6, 93bitr3g 211 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  wal 1241  [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-bndl 1399  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: (None)
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