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Theorem sbal2 2460
Description: Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 3-Oct-2018.)
Assertion
Ref Expression
sbal2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbal2
StepHypRef Expression
1 sb4b 2357 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
21adantl 482 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
3 nfnae 2317 . . . . . 6 𝑥 ¬ ∀𝑦 𝑦 = 𝑧
4 sb4b 2357 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧𝜑)))
53, 4albid 2088 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥𝑦(𝑦 = 𝑧𝜑)))
6 alcom 2034 . . . . . 6 (∀𝑥𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦𝑥(𝑦 = 𝑧𝜑))
7 nfnae 2317 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
8 nfeqf1 2298 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
9 19.21t 2071 . . . . . . . 8 (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑦 = 𝑧𝜑) ↔ (𝑦 = 𝑧 → ∀𝑥𝜑)))
108, 9syl 17 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑦 = 𝑧𝜑) ↔ (𝑦 = 𝑧 → ∀𝑥𝜑)))
117, 10albid 2088 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥(𝑦 = 𝑧𝜑) ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
126, 11syl5bb 272 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
135, 12sylan9bbr 736 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
142, 13bitr4d 271 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
1514ex 450 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)))
16 sbid 2111 . . . 4 ([𝑦 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥𝜑)
17 drsb2 2377 . . . 4 (∀𝑦 𝑦 = 𝑧 → ([𝑦 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
1816, 17syl5bbr 274 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
19 sbid 2111 . . . . 5 ([𝑦 / 𝑦]𝜑𝜑)
20 drsb2 2377 . . . . 5 (∀𝑦 𝑦 = 𝑧 → ([𝑦 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
2119, 20syl5bbr 274 . . . 4 (∀𝑦 𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
2221dral2 2323 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
2318, 22bitr3d 270 . 2 (∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
2415, 23pm2.61d2 172 1 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478  wnf 1705  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by:  2sb5ndVD  38626  2sb5ndALT  38648
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