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Theorem equs4 1959
 Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
Assertion
Ref Expression
equs4 (x(x = yφ) → x(x = y φ))

Proof of Theorem equs4
StepHypRef Expression
1 a9e 1951 . . 3 x x = y
2 19.29 1596 . . 3 ((x(x = yφ) x x = y) → x((x = yφ) x = y))
31, 2mpan2 652 . 2 (x(x = yφ) → x((x = yφ) x = y))
4 ancl 529 . . . 4 ((x = yφ) → (x = y → (x = y φ)))
54imp 418 . . 3 (((x = yφ) x = y) → (x = y φ))
65eximi 1576 . 2 (x((x = yφ) x = y) → x(x = y φ))
73, 6syl 15 1 (x(x = yφ) → x(x = y φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  equs45f  1989  sb2  2023
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