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Theorem sbco2 2307
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 2130 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑))
2 sbequ 2268 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
31, 2bitr3d 268 . . 3 (𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
43sps 1996 . 2 (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
5 nfnae 2210 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
6 sbco2.1 . . . 4 𝑧𝜑
76nfsb4 2282 . . 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 2301 . 2 (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
104, 9pm2.61i 174 1 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wal 1472  wnf 1698  [wsb 1830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1699  df-sb 1831
This theorem is referenced by:  sbco2d  2308  equsb3ALT  2325  elsb3  2326  elsb4  2327  sb7f  2345  sbco4lem  2357  sbco4  2358  eqsb3  2619  clelsb3  2620  cbvab  2637  sbralie  3064  sbcco  3329  clelsb3f  28496  bj-clelsb3  31877
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