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Theorem sbal1 2565
Description: Obsolete version of sbal 2156 as of 13-Jul-2023. A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 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 2492 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
2 nfnae 2448 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑧
3 nfeqf2 2386 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧)
4 19.21t 2196 . . . . . . . 8 (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑦 = 𝑧𝜑) ↔ (𝑦 = 𝑧 → ∀𝑥𝜑)))
54bicomd 224 . . . . . . 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 2214 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑)))
81, 7sylan9bbr 511 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑)))
9 nfnae 2448 . . . . . . 7 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
10 sb4b 2492 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧𝜑)))
119, 10albid 2214 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥𝑦(𝑦 = 𝑧𝜑)))
12 alcom 2153 . . . . . 6 (∀𝑥𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
1311, 12syl6bb 288 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑)))
1413adantl 482 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑)))
158, 14bitr4d 283 . . 3 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
1615ex 413 . 2 (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)))
17 sbequ12 2243 . . . 4 (𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
1817sps 2174 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
19 sbequ12 2243 . . . . 5 (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
2019sps 2174 . . . 4 (∀𝑦 𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
2120dral2 2452 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
2218, 21bitr3d 282 . 2 (∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
2316, 22pm2.61d2 182 1 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526  wnf 1775  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061
This theorem is referenced by:  sbalOLD  2568
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