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Theorem sbco2 2413
Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.)
Hypothesis
Ref Expression
sbco2.1 𝑧𝜑
Assertion
Ref Expression
sbco2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbco2
StepHypRef Expression
1 sbequ12 2109 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑))
2 sbequ 2374 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
31, 2bitr3d 270 . . 3 (𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
43sps 2053 . 2 (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
5 nfnae 2316 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
6 sbco2.1 . . . 4 𝑧𝜑
76nfsb4 2388 . . 3 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
82a1i 11 . . 3 (¬ ∀𝑧 𝑧 = 𝑦 → (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)))
95, 7, 8sbied 2407 . 2 (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
104, 9pm2.61i 176 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1479  wnf 1706  [wsb 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879
This theorem is referenced by:  sbco2d  2414  equsb3ALT  2431  elsb3  2432  elsb4  2433  sb7f  2451  sbco4lem  2463  sbco4  2464  eqsb3  2726  clelsb3  2727  cbvab  2744  sbralie  3179  sbcco  3452  clelsb3f  29292  bj-clelsb3  32823
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