MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbco2d Structured version   Visualization version   GIF version

Theorem sbco2d 2444
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 2105 . . . 4 𝑧(𝜑𝜓)
43sbco2 2443 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥](𝜑𝜓))
5 sbco2d.1 . . . . . 6 𝑥𝜑
65sbrim 2424 . . . . 5 ([𝑧 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓))
76sbbii 1944 . . . 4 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓))
81sbrim 2424 . . . 4 ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
97, 8bitri 264 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
105sbrim 2424 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
114, 9, 103bitr3i 290 . 2 ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
1211pm5.74ri 261 1 (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1748  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  sbco3  2445
  Copyright terms: Public domain W3C validator