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Theorem sbco3 2532
 Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.) (New usage is discouraged.)
Assertion
Ref Expression
sbco3 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Proof of Theorem sbco3
StepHypRef Expression
1 drsb1 2513 . . 3 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
2 nfae 2444 . . . 4 𝑥𝑥 𝑥 = 𝑦
3 sbequ12a 2253 . . . . 5 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))
43sps 2182 . . . 4 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))
52, 4sbbid 2244 . . 3 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑))
61, 5bitr3d 284 . 2 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑))
7 nfnae 2445 . . . 4 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
8 nfnae 2445 . . . 4 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
9 nfsb2 2501 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
107, 8, 9sbco2d 2531 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
11 sbco 2526 . . . 4 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑)
1211sbbii 2081 . . 3 ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
1310, 12bitr3di 289 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑))
146, 13pm2.61i 185 1 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1536  [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:  sbcom  2533
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