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Theorem sbal1 1975
 Description: A theorem used in elimination of disjoint variable conditions on 𝑥, 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 1973 . . . 4 ([𝑤 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑤 / 𝑦]𝜑)
21sbbii 1738 . . 3 ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑤]∀𝑥[𝑤 / 𝑦]𝜑)
3 sbal1yz 1974 . . 3 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑤]∀𝑥[𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑))
42, 3syl5bb 191 . 2 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑))
5 ax-17 1506 . . 3 (∀𝑥𝜑 → ∀𝑤𝑥𝜑)
65sbco2vh 1916 . 2 ([𝑧 / 𝑤][𝑤 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑)
7 ax-17 1506 . . . 4 (𝜑 → ∀𝑤𝜑)
87sbco2vh 1916 . . 3 ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑)
98albii 1446 . 2 (∀𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
104, 6, 93bitr3g 221 1 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  ∀wal 1329  [wsb 1735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736 This theorem is referenced by: (None)
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