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Theorem equs5or 1727
Description: Lemma used in proofs of substitution properties. Like equs5 1726 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)
Assertion
Ref Expression
equs5or (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem equs5or
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 a9e 1602 . 2 𝑧 𝑧 = 𝑦
2 dveeq2or 1713 . . . . . 6 (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦)
3 nfnf1 1452 . . . . . . . . . . 11 𝑥𝑥 𝑧 = 𝑦
43nfri 1428 . . . . . . . . . 10 (Ⅎ𝑥 𝑧 = 𝑦 → ∀𝑥𝑥 𝑧 = 𝑦)
5 ax11v 1724 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
6 equequ2 1615 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
76adantl 266 . . . . . . . . . . . . . 14 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → (𝑥 = 𝑧𝑥 = 𝑦))
8 nfr 1427 . . . . . . . . . . . . . . . . 17 (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
98imp 119 . . . . . . . . . . . . . . . 16 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → ∀𝑥 𝑧 = 𝑦)
10 hba1 1449 . . . . . . . . . . . . . . . . 17 (∀𝑥 𝑧 = 𝑦 → ∀𝑥𝑥 𝑧 = 𝑦)
116imbi1d 224 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
1211sps 1446 . . . . . . . . . . . . . . . . 17 (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
1310, 12albidh 1385 . . . . . . . . . . . . . . . 16 (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
149, 13syl 14 . . . . . . . . . . . . . . 15 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → (∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
1514imbi2d 223 . . . . . . . . . . . . . 14 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → ((𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
167, 15imbi12d 227 . . . . . . . . . . . . 13 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
175, 16mpbii 140 . . . . . . . . . . . 12 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
1817ex 112 . . . . . . . . . . 11 (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
1918imp4a 335 . . . . . . . . . 10 (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
204, 19alrimih 1374 . . . . . . . . 9 (Ⅎ𝑥 𝑧 = 𝑦 → ∀𝑥(𝑧 = 𝑦 → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
21 19.21t 1490 . . . . . . . . 9 (Ⅎ𝑥 𝑧 = 𝑦 → (∀𝑥(𝑧 = 𝑦 → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))) ↔ (𝑧 = 𝑦 → ∀𝑥((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))))
2220, 21mpbid 139 . . . . . . . 8 (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
23 hba1 1449 . . . . . . . . 9 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝑥(𝑥 = 𝑦𝜑))
242319.23h 1403 . . . . . . . 8 (∀𝑥((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
2522, 24syl6ib 154 . . . . . . 7 (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
2625orim2i 688 . . . . . 6 ((∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦) → (∀𝑥 𝑥 = 𝑦 ∨ (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))))
272, 26ax-mp 7 . . . . 5 (∀𝑥 𝑥 = 𝑦 ∨ (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
28 pm2.76 732 . . . . 5 ((∀𝑥 𝑥 = 𝑦 ∨ (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))) → ((∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))))
2927, 28ax-mp 7 . . . 4 ((∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
3029olcs 665 . . 3 (𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
3130exlimiv 1505 . 2 (∃𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
321, 31ax-mp 7 1 (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 639  wal 1257  wnf 1365  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  sb4or  1730
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