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Theorem equs5or 1711
 Description: Lemma used in proofs of substitution properties. Like equs5 1710 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 1586 . 2 𝑧 𝑧 = 𝑦
2 dveeq2or 1697 . . . . . 6 (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦)
3 nfnf1 1436 . . . . . . . . . . 11 𝑥𝑥 𝑧 = 𝑦
43nfri 1412 . . . . . . . . . 10 (Ⅎ𝑥 𝑧 = 𝑦 → ∀𝑥𝑥 𝑧 = 𝑦)
5 ax11v 1708 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
6 equequ2 1599 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
76adantl 262 . . . . . . . . . . . . . 14 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → (𝑥 = 𝑧𝑥 = 𝑦))
8 nfr 1411 . . . . . . . . . . . . . . . . 17 (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
98imp 115 . . . . . . . . . . . . . . . 16 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → ∀𝑥 𝑧 = 𝑦)
10 hba1 1433 . . . . . . . . . . . . . . . . 17 (∀𝑥 𝑧 = 𝑦 → ∀𝑥𝑥 𝑧 = 𝑦)
116imbi1d 220 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
1211sps 1430 . . . . . . . . . . . . . . . . 17 (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
1310, 12albidh 1369 . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . 14 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → ((𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
167, 15imbi12d 223 . . . . . . . . . . . . 13 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
175, 16mpbii 136 . . . . . . . . . . . 12 ((Ⅎ𝑥 𝑧 = 𝑦𝑧 = 𝑦) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
1817ex 108 . . . . . . . . . . 11 (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
1918imp4a 331 . . . . . . . . . 10 (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
204, 19alrimih 1358 . . . . . . . . 9 (Ⅎ𝑥 𝑧 = 𝑦 → ∀𝑥(𝑧 = 𝑦 → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
21 19.21t 1474 . . . . . . . . 9 (Ⅎ𝑥 𝑧 = 𝑦 → (∀𝑥(𝑧 = 𝑦 → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))) ↔ (𝑧 = 𝑦 → ∀𝑥((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))))
2220, 21mpbid 135 . . . . . . . 8 (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
23 hba1 1433 . . . . . . . . 9 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝑥(𝑥 = 𝑦𝜑))
242319.23h 1387 . . . . . . . 8 (∀𝑥((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
2522, 24syl6ib 150 . . . . . . 7 (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
2625orim2i 678 . . . . . 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 pm2.76 721 . . . . 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 655 . . 3 (𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))))
3130exlimiv 1489 . 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 97   ↔ wb 98   ∨ wo 629  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  sb4or  1714
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