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Theorem sbal1 2548
 Description: Check out sbal 2163 for a version not dependent on ax-13 2379. A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑧 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
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
StepHypRef Expression
1 sb4b 2488 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
2 nfnae 2445 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑧
3 nfeqf2 2384 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧)
4 19.21t 2204 . . . . . . . 8 (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑦 = 𝑧𝜑) ↔ (𝑦 = 𝑧 → ∀𝑥𝜑)))
54bicomd 226 . . . . . . 7 (Ⅎ𝑥 𝑦 = 𝑧 → ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑥(𝑦 = 𝑧𝜑)))
63, 5syl 17 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑧 → ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑥(𝑦 = 𝑧𝜑)))
72, 6albid 2222 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑)))
81, 7sylan9bbr 514 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑)))
9 nfnae 2445 . . . . . . 7 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
10 sb4b 2488 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧𝜑)))
119, 10albid 2222 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥𝑦(𝑦 = 𝑧𝜑)))
12 alcom 2160 . . . . . 6 (∀𝑥𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
1311, 12syl6bb 290 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑)))
1413adantl 485 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑)))
158, 14bitr4d 285 . . 3 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
1615ex 416 . 2 (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)))
17 sbequ12 2250 . . . 4 (𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
1817sps 2182 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
19 sbequ12 2250 . . . . 5 (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
2019sps 2182 . . . 4 (∀𝑦 𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
2120dral2 2449 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
2218, 21bitr3d 284 . 2 (∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
2316, 22pm2.61d2 184 1 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  Ⅎwnf 1785  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by:  sbalOLD  2551
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