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Theorem equsex 1962
 Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
equsex.1 xψ
equsex.2 (x = y → (φψ))
Assertion
Ref Expression
equsex (x(x = y φ) ↔ ψ)

Proof of Theorem equsex
StepHypRef Expression
1 exnal 1574 . 2 (x ¬ (x = y → ¬ φ) ↔ ¬ x(x = y → ¬ φ))
2 df-an 360 . . 3 ((x = y φ) ↔ ¬ (x = y → ¬ φ))
32exbii 1582 . 2 (x(x = y φ) ↔ x ¬ (x = y → ¬ φ))
4 equsex.1 . . . . 5 xψ
54nfn 1793 . . . 4 x ¬ ψ
6 equsex.2 . . . . 5 (x = y → (φψ))
76notbid 285 . . . 4 (x = y → (¬ φ ↔ ¬ ψ))
85, 7equsal 1960 . . 3 (x(x = y → ¬ φ) ↔ ¬ ψ)
98con2bii 322 . 2 (ψ ↔ ¬ x(x = y → ¬ φ))
101, 3, 93bitr4i 268 1 (x(x = y φ) ↔ ψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  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:  equsexh  1963  cleljustALT  2015  sb56  2098  sb10f  2122
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