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Theorem equsal 1960
 Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
equsal.1 xψ
equsal.2 (x = y → (φψ))
Assertion
Ref Expression
equsal (x(x = yφ) ↔ ψ)

Proof of Theorem equsal
StepHypRef Expression
1 equsal.2 . . . . 5 (x = y → (φψ))
2 equsal.1 . . . . . 6 xψ
3219.3 1785 . . . . 5 (xψψ)
41, 3syl6bbr 254 . . . 4 (x = y → (φxψ))
54pm5.74i 236 . . 3 ((x = yφ) ↔ (x = yxψ))
65albii 1566 . 2 (x(x = yφ) ↔ x(x = yxψ))
72nfri 1762 . . . . 5 (ψxψ)
87a1d 22 . . . 4 (ψ → (x = yxψ))
92, 8alrimi 1765 . . 3 (ψx(x = yxψ))
10 ax9o 1950 . . 3 (x(x = yxψ) → ψ)
119, 10impbii 180 . 2 (ψx(x = yxψ))
126, 11bitr4i 243 1 (x(x = yφ) ↔ ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  equsalh  1961  equsex  1962  dvelimf  1997  sb6x  2029  fun11  5159
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