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Theorem sbco2d 2547
Description: A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
sbco2d.1 𝑥𝜑
sbco2d.2 𝑧𝜑
sbco2d.3 (𝜑 → Ⅎ𝑧𝜓)
Assertion
Ref Expression
sbco2d (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))

Proof of Theorem sbco2d
StepHypRef Expression
1 sbco2d.2 . . . . 5 𝑧𝜑
2 sbco2d.3 . . . . 5 (𝜑 → Ⅎ𝑧𝜓)
31, 2nfim1 2189 . . . 4 𝑧(𝜑𝜓)
43sbco2 2546 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥](𝜑𝜓))
5 sbco2d.1 . . . . . 6 𝑥𝜑
65sbrim 2304 . . . . 5 ([𝑧 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓))
76sbbii 2072 . . . 4 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓))
81sbrim 2304 . . . 4 ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
97, 8bitri 276 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
105sbrim 2304 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
114, 9, 103bitr3i 302 . 2 ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
1211pm5.74ri 273 1 (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  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:  sbco3  2548  wl-clelsb3df  34744
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