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Theorem sbequi 1702
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)
Assertion
Ref Expression
sbequi (x = y → ([x / z]φ → [y / z]φ))

Proof of Theorem sbequi
StepHypRef Expression
1 nfsb2or 1700 . . . 4 (z z = x z[x / z]φ)
2 nfr 1392 . . . . . 6 (Ⅎz[x / z]φ → ([x / z]φz[x / z]φ))
3 equvini 1623 . . . . . . 7 (x = yz(x = z z = y))
4 stdpc7 1635 . . . . . . . . 9 (x = z → ([x / z]φφ))
5 sbequ1 1633 . . . . . . . . 9 (z = y → (φ → [y / z]φ))
64, 5sylan9 391 . . . . . . . 8 ((x = z z = y) → ([x / z]φ → [y / z]φ))
76eximi 1473 . . . . . . 7 (z(x = z z = y) → z([x / z]φ → [y / z]φ))
8 19.35-1 1497 . . . . . . 7 (z([x / z]φ → [y / z]φ) → (z[x / z]φz[y / z]φ))
93, 7, 83syl 17 . . . . . 6 (x = y → (z[x / z]φz[y / z]φ))
102, 9syl9 66 . . . . 5 (Ⅎz[x / z]φ → (x = y → ([x / z]φz[y / z]φ)))
1110orim2i 665 . . . 4 ((z z = x z[x / z]φ) → (z z = x (x = y → ([x / z]φz[y / z]φ))))
121, 11ax-mp 7 . . 3 (z z = x (x = y → ([x / z]φz[y / z]φ)))
13 nfsb2or 1700 . . . . 5 (z z = y z[y / z]φ)
14 19.9t 1515 . . . . . . 7 (Ⅎz[y / z]φ → (z[y / z]φ ↔ [y / z]φ))
1514biimpd 132 . . . . . 6 (Ⅎz[y / z]φ → (z[y / z]φ → [y / z]φ))
1615orim2i 665 . . . . 5 ((z z = y z[y / z]φ) → (z z = y (z[y / z]φ → [y / z]φ)))
1713, 16ax-mp 7 . . . 4 (z z = y (z[y / z]φ → [y / z]φ))
18 ax-1 5 . . . . 5 ((z[y / z]φ → [y / z]φ) → (x = y → (z[y / z]φ → [y / z]φ)))
1918orim2i 665 . . . 4 ((z z = y (z[y / z]φ → [y / z]φ)) → (z z = y (x = y → (z[y / z]φ → [y / z]φ))))
2017, 19ax-mp 7 . . 3 (z z = y (x = y → (z[y / z]φ → [y / z]φ)))
2112, 20sbequilem 1701 . 2 (z z = x (z z = y (x = y → ([x / z]φ → [y / z]φ))))
22 sbequ2 1634 . . . . . . 7 (z = x → ([x / z]φφ))
2322sps 1412 . . . . . 6 (z z = x → ([x / z]φφ))
2423adantr 261 . . . . 5 ((z z = x x = y) → ([x / z]φφ))
25 sbequ1 1633 . . . . . 6 (x = y → (φ → [y / x]φ))
26 drsb1 1662 . . . . . . . 8 (x x = z → ([y / x]φ ↔ [y / z]φ))
2726biimpd 132 . . . . . . 7 (x x = z → ([y / x]φ → [y / z]φ))
2827alequcoms 1390 . . . . . 6 (z z = x → ([y / x]φ → [y / z]φ))
2925, 28sylan9r 392 . . . . 5 ((z z = x x = y) → (φ → [y / z]φ))
3024, 29syld 40 . . . 4 ((z z = x x = y) → ([x / z]φ → [y / z]φ))
3130ex 108 . . 3 (z z = x → (x = y → ([x / z]φ → [y / z]φ)))
32 drsb1 1662 . . . . . . . . 9 (z z = y → ([x / z]φ ↔ [x / y]φ))
3332biimpd 132 . . . . . . . 8 (z z = y → ([x / z]φ → [x / y]φ))
34 stdpc7 1635 . . . . . . . 8 (x = y → ([x / y]φφ))
3533, 34sylan9 391 . . . . . . 7 ((z z = y x = y) → ([x / z]φφ))
365sps 1412 . . . . . . . 8 (z z = y → (φ → [y / z]φ))
3736adantr 261 . . . . . . 7 ((z z = y x = y) → (φ → [y / z]φ))
3835, 37syld 40 . . . . . 6 ((z z = y x = y) → ([x / z]φ → [y / z]φ))
3938ex 108 . . . . 5 (z z = y → (x = y → ([x / z]φ → [y / z]φ)))
4039orim1i 664 . . . 4 ((z z = y (x = y → ([x / z]φ → [y / z]φ))) → ((x = y → ([x / z]φ → [y / z]φ)) (x = y → ([x / z]φ → [y / z]φ))))
41 pm1.2 660 . . . 4 (((x = y → ([x / z]φ → [y / z]φ)) (x = y → ([x / z]φ → [y / z]φ))) → (x = y → ([x / z]φ → [y / z]φ)))
4240, 41syl 14 . . 3 ((z z = y (x = y → ([x / z]φ → [y / z]φ))) → (x = y → ([x / z]φ → [y / z]φ)))
4331, 42jaoi 623 . 2 ((z z = x (z z = y (x = y → ([x / z]φ → [y / z]φ)))) → (x = y → ([x / z]φ → [y / z]φ)))
4421, 43ax-mp 7 1 (x = y → ([x / z]φ → [y / z]φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616  wal 1226  wnf 1329  wex 1362  [wsb 1627
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628
This theorem is referenced by:  sbequ  1703
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