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Theorem equs45f 1989
 Description: Two ways of expressing substitution when y is not free in φ. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
equs45f.1 yφ
Assertion
Ref Expression
equs45f (x(x = y φ) ↔ x(x = yφ))

Proof of Theorem equs45f
StepHypRef Expression
1 equs45f.1 . . . . . 6 yφ
21nfri 1762 . . . . 5 (φyφ)
32anim2i 552 . . . 4 ((x = y φ) → (x = y yφ))
43eximi 1576 . . 3 (x(x = y φ) → x(x = y yφ))
5 equs5a 1887 . . 3 (x(x = y yφ) → x(x = yφ))
64, 5syl 15 . 2 (x(x = y φ) → x(x = yφ))
7 equs4 1959 . 2 (x(x = yφ) → x(x = y φ))
86, 7impbii 180 1 (x(x = y φ) ↔ x(x = yφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544 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:  sb5f  2040
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