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Theorem sb4or 1761
Description: One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1760 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
Assertion
Ref Expression
sb4or (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem sb4or
StepHypRef Expression
1 equs5or 1758 . 2 (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
2 nfe1 1430 . . . . . 6 𝑥𝑥(𝑥 = 𝑦𝜑)
3 nfa1 1479 . . . . . 6 𝑥𝑥(𝑥 = 𝑦𝜑)
42, 3nfim 1509 . . . . 5 𝑥(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
54nfri 1457 . . . 4 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑥(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
6 sb1 1696 . . . . 5 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
76imim1i 59 . . . 4 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)) → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
85, 7alrimih 1403 . . 3 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
98orim2i 713 . 2 ((∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))) → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
101, 9ax-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  wo 664  wal 1287  wex 1426  [wsb 1692
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  sb4bor  1763  nfsb2or  1765
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