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Theorem ceqex 2635
Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
Assertion
Ref Expression
ceqex (x = A → (φx(x = A φ)))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem ceqex
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 19.8a 1451 . . 3 (x = Ax x = A)
2 isset 2526 . . 3 (A V ↔ x x = A)
31, 2sylibr 137 . 2 (x = AA V)
4 eqeq2 2018 . . . 4 (y = A → (x = yx = A))
54anbi1d 438 . . . . . 6 (y = A → ((x = y φ) ↔ (x = A φ)))
65exbidv 1675 . . . . 5 (y = A → (x(x = y φ) ↔ x(x = A φ)))
76bibi2d 221 . . . 4 (y = A → ((φx(x = y φ)) ↔ (φx(x = A φ))))
84, 7imbi12d 223 . . 3 (y = A → ((x = y → (φx(x = y φ))) ↔ (x = A → (φx(x = A φ)))))
9 19.8a 1451 . . . . 5 ((x = y φ) → x(x = y φ))
109ex 108 . . . 4 (x = y → (φx(x = y φ)))
11 vex 2525 . . . . . 6 y V
1211alexeq 2634 . . . . 5 (x(x = yφ) ↔ x(x = y φ))
13 sp 1370 . . . . . 6 (x(x = yφ) → (x = yφ))
1413com12 27 . . . . 5 (x = y → (x(x = yφ) → φ))
1512, 14syl5bir 142 . . . 4 (x = y → (x(x = y φ) → φ))
1610, 15impbid 120 . . 3 (x = y → (φx(x = y φ)))
178, 16vtoclg 2577 . 2 (A V → (x = A → (φx(x = A φ))))
183, 17mpcom 32 1 (x = A → (φx(x = A φ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1217   = wceq 1219  wex 1350   wcel 1362  Vcvv 2522
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 613  ax-5 1305  ax-7 1306  ax-gen 1307  ax-ie1 1351  ax-ie2 1352  ax-8 1364  ax-10 1365  ax-11 1366  ax-i12 1367  ax-bnd 1368  ax-4 1369  ax-17 1388  ax-i9 1392  ax-ial 1396  ax-i5r 1397  ax-ext 1991
This theorem depends on definitions:  df-bi 110  df-tru 1222  df-nf 1319  df-sb 1615  df-clab 1996  df-cleq 2002  df-clel 2005  df-nfc 2136  df-v 2524
This theorem is referenced by:  ceqsexg  2636  sbc6g  2752
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