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Theorem sbco2d 1882
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
sbco2d.1 (𝜑 → ∀𝑥𝜑)
sbco2d.2 (𝜑 → ∀𝑧𝜑)
sbco2d.3 (𝜑 → (𝜓 → ∀𝑧𝜓))
Assertion
Ref Expression
sbco2d (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))

Proof of Theorem sbco2d
StepHypRef Expression
1 sbco2d.2 . . . . 5 (𝜑 → ∀𝑧𝜑)
2 sbco2d.3 . . . . 5 (𝜑 → (𝜓 → ∀𝑧𝜓))
31, 2hbim1 1503 . . . 4 ((𝜑𝜓) → ∀𝑧(𝜑𝜓))
43sbco2h 1880 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥](𝜑𝜓))
5 sbco2d.1 . . . . . 6 (𝜑 → ∀𝑥𝜑)
65sbrim 1872 . . . . 5 ([𝑧 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓))
76sbbii 1689 . . . 4 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓))
81sbrim 1872 . . . 4 ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
97, 8bitri 182 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
105sbrim 1872 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
114, 9, 103bitr3i 208 . 2 ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
1211pm5.74ri 179 1 (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by: (None)
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