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Theorem equs5or 1708
Description: Lemma used in proofs of substitution properties. Like equs5 1707 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 = y φ) → x(x = yφ)))

Proof of Theorem equs5or
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 a9e 1583 . 2 z z = y
2 dveeq2or 1694 . . . . . 6 (x x = y x z = y)
3 nfnf1 1433 . . . . . . . . . . 11 xx z = y
43nfri 1409 . . . . . . . . . 10 (Ⅎx z = yxx z = y)
5 ax11v 1705 . . . . . . . . . . . . 13 (x = z → (φx(x = zφ)))
6 equequ2 1596 . . . . . . . . . . . . . . 15 (z = y → (x = zx = y))
76adantl 262 . . . . . . . . . . . . . 14 ((Ⅎx z = y z = y) → (x = zx = y))
8 nfr 1408 . . . . . . . . . . . . . . . . 17 (Ⅎx z = y → (z = yx z = y))
98imp 115 . . . . . . . . . . . . . . . 16 ((Ⅎx z = y z = y) → x z = y)
10 hba1 1430 . . . . . . . . . . . . . . . . 17 (x z = yxx z = y)
116imbi1d 220 . . . . . . . . . . . . . . . . . 18 (z = y → ((x = zφ) ↔ (x = yφ)))
1211sps 1427 . . . . . . . . . . . . . . . . 17 (x z = y → ((x = zφ) ↔ (x = yφ)))
1310, 12albidh 1366 . . . . . . . . . . . . . . . 16 (x z = y → (x(x = zφ) ↔ x(x = yφ)))
149, 13syl 14 . . . . . . . . . . . . . . 15 ((Ⅎx z = y z = y) → (x(x = zφ) ↔ x(x = yφ)))
1514imbi2d 219 . . . . . . . . . . . . . 14 ((Ⅎx z = y z = y) → ((φx(x = zφ)) ↔ (φx(x = yφ))))
167, 15imbi12d 223 . . . . . . . . . . . . 13 ((Ⅎx z = y z = y) → ((x = z → (φx(x = zφ))) ↔ (x = y → (φx(x = yφ)))))
175, 16mpbii 136 . . . . . . . . . . . 12 ((Ⅎx z = y z = y) → (x = y → (φx(x = yφ))))
1817ex 108 . . . . . . . . . . 11 (Ⅎx z = y → (z = y → (x = y → (φx(x = yφ)))))
1918imp4a 331 . . . . . . . . . 10 (Ⅎx z = y → (z = y → ((x = y φ) → x(x = yφ))))
204, 19alrimih 1355 . . . . . . . . 9 (Ⅎx z = yx(z = y → ((x = y φ) → x(x = yφ))))
21 19.21t 1471 . . . . . . . . 9 (Ⅎx z = y → (x(z = y → ((x = y φ) → x(x = yφ))) ↔ (z = yx((x = y φ) → x(x = yφ)))))
2220, 21mpbid 135 . . . . . . . 8 (Ⅎx z = y → (z = yx((x = y φ) → x(x = yφ))))
23 hba1 1430 . . . . . . . . 9 (x(x = yφ) → xx(x = yφ))
242319.23h 1384 . . . . . . . 8 (x((x = y φ) → x(x = yφ)) ↔ (x(x = y φ) → x(x = yφ)))
2522, 24syl6ib 150 . . . . . . 7 (Ⅎx z = y → (z = y → (x(x = y φ) → x(x = yφ))))
2625orim2i 677 . . . . . 6 ((x x = y x z = y) → (x x = y (z = y → (x(x = y φ) → x(x = yφ)))))
272, 26ax-mp 7 . . . . 5 (x x = y (z = y → (x(x = y φ) → x(x = yφ))))
28 pm2.76 720 . . . . 5 ((x x = y (z = y → (x(x = y φ) → x(x = yφ)))) → ((x x = y z = y) → (x x = y (x(x = y φ) → x(x = yφ)))))
2927, 28ax-mp 7 . . . 4 ((x x = y z = y) → (x x = y (x(x = y φ) → x(x = yφ))))
3029olcs 654 . . 3 (z = y → (x x = y (x(x = y φ) → x(x = yφ))))
3130exlimiv 1486 . 2 (z z = y → (x x = y (x(x = y φ) → x(x = yφ))))
321, 31ax-mp 7 1 (x x = y (x(x = y φ) → x(x = yφ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628  wal 1240  wnf 1346  wex 1378
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  sb4or  1711
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