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Theorem alexeq 2968
 Description: Two ways to express substitution of A for x in φ. (Contributed by NM, 2-Mar-1995.)
Hypothesis
Ref Expression
alexeq.1 A V
Assertion
Ref Expression
alexeq (x(x = Aφ) ↔ x(x = A φ))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem alexeq
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 alexeq.1 . . 3 A V
2 eqeq2 2362 . . . . 5 (y = A → (x = yx = A))
32anbi1d 685 . . . 4 (y = A → ((x = y φ) ↔ (x = A φ)))
43exbidv 1626 . . 3 (y = A → (x(x = y φ) ↔ x(x = A φ)))
52imbi1d 308 . . . 4 (y = A → ((x = yφ) ↔ (x = Aφ)))
65albidv 1625 . . 3 (y = A → (x(x = yφ) ↔ x(x = Aφ)))
7 sb56 2098 . . 3 (x(x = y φ) ↔ x(x = yφ))
81, 4, 6, 7vtoclb 2912 . 2 (x(x = A φ) ↔ x(x = Aφ))
98bicomi 193 1 (x(x = Aφ) ↔ x(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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  ceqex  2969
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