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Theorem sbco2d 2507
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 2229 . . . 4 𝑧(𝜑𝜓)
43sbco2 2506 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥](𝜑𝜓))
5 sbco2d.1 . . . . . 6 𝑥𝜑
65sbrim 2487 . . . . 5 ([𝑧 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓))
76sbbii 2069 . . . 4 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓))
81sbrim 2487 . . . 4 ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
97, 8bitri 266 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
105sbrim 2487 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
114, 9, 103bitr3i 292 . 2 ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
1211pm5.74ri 263 1 (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wnf 1878  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063
This theorem is referenced by:  sbco3  2508
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