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Theorem ceqex 2969
 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 1756 . . 3 (x = Ax x = A)
2 isset 2863 . . 3 (A V ↔ x x = A)
31, 2sylibr 203 . 2 (x = AA V)
4 eqeq2 2362 . . . 4 (y = A → (x = yx = A))
54anbi1d 685 . . . . . 6 (y = A → ((x = y φ) ↔ (x = A φ)))
65exbidv 1626 . . . . 5 (y = A → (x(x = y φ) ↔ x(x = A φ)))
76bibi2d 309 . . . 4 (y = A → ((φx(x = y φ)) ↔ (φx(x = A φ))))
84, 7imbi12d 311 . . 3 (y = A → ((x = y → (φx(x = y φ))) ↔ (x = A → (φx(x = A φ)))))
9 19.8a 1756 . . . . 5 ((x = y φ) → x(x = y φ))
109ex 423 . . . 4 (x = y → (φx(x = y φ)))
11 vex 2862 . . . . . 6 y V
1211alexeq 2968 . . . . 5 (x(x = yφ) ↔ x(x = y φ))
13 sp 1747 . . . . . 6 (x(x = yφ) → (x = yφ))
1413com12 27 . . . . 5 (x = y → (x(x = yφ) → φ))
1512, 14syl5bir 209 . . . 4 (x = y → (x(x = y φ) → φ))
1610, 15impbid 183 . . 3 (x = y → (φx(x = y φ)))
178, 16vtoclg 2914 . 2 (A V → (x = A → (φx(x = A φ))))
183, 17mpcom 32 1 (x = A → (φx(x = A φ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by:  ceqsexg  2970  sbc6g  3071
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