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Theorem sbco2vd 1940
 Description: Version of sbco2d 1939 with a distinct variable constraint between 𝑥 and 𝑧. (Contributed by Jim Kingdon, 19-Feb-2018.)
Hypotheses
Ref Expression
sbco2vd.1 (𝜑 → ∀𝑥𝜑)
sbco2vd.2 (𝜑 → ∀𝑧𝜑)
sbco2vd.3 (𝜑 → (𝜓 → ∀𝑧𝜓))
Assertion
Ref Expression
sbco2vd (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem sbco2vd
StepHypRef Expression
1 sbco2vd.2 . . . . 5 (𝜑 → ∀𝑧𝜑)
2 sbco2vd.3 . . . . 5 (𝜑 → (𝜓 → ∀𝑧𝜓))
31, 2hbim1 1549 . . . 4 ((𝜑𝜓) → ∀𝑧(𝜑𝜓))
43sbco2vh 1918 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑥](𝜑𝜓))
5 sbco2vd.1 . . . . . 6 (𝜑 → ∀𝑥𝜑)
65sbrim 1929 . . . . 5 ([𝑧 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓))
76sbbii 1738 . . . 4 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓))
81sbrim 1929 . . . 4 ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
97, 8bitri 183 . . 3 ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
105sbrim 1929 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
114, 9, 103bitr3i 209 . 2 ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
1211pm5.74ri 180 1 (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  [wsb 1735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736 This theorem is referenced by: (None)
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