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Theorem sb4or 1843
Description: One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1842 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 1840 . 2 (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
2 nfe1 1506 . . . . . 6 𝑥𝑥(𝑥 = 𝑦𝜑)
3 nfa1 1551 . . . . . 6 𝑥𝑥(𝑥 = 𝑦𝜑)
42, 3nfim 1582 . . . . 5 𝑥(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
54nfri 1529 . . . 4 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑥(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
6 sb1 1776 . . . . 5 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
76imim1i 60 . . . 4 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)) → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
85, 7alrimih 1479 . . 3 ((∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
98orim2i 762 . 2 ((∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))) → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
101, 9ax-mp 5 1 (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709  wal 1361  wex 1502  [wsb 1772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773
This theorem is referenced by:  sb4bor  1845  nfsb2or  1847
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