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Theorem sb4bor 1757
Description: Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)
Assertion
Ref Expression
sb4bor (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem sb4bor
StepHypRef Expression
1 sb4or 1755 . 2 (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 sb2 1691 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
3 df-bi 115 . . . . . 6 ((([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)) → (([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)) ∧ (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))) ∧ ((([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)) ∧ (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))))
43simpri 111 . . . . 5 ((([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)) ∧ (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
52, 4mpan2 416 . . . 4 (([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)) → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
65alimi 1385 . . 3 (∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑥([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
76orim2i 711 . 2 ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))) → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))))
81, 7ax-mp 7 1 (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  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-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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