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Theorem sbft 2025
 Description: Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.)
Assertion
Ref Expression
sbft (Ⅎxφ → ([y / x]φφ))

Proof of Theorem sbft
StepHypRef Expression
1 sb1 1651 . . 3 ([y / x]φx(x = y φ))
2 simpr 447 . . . . 5 ((x = y φ) → φ)
32ax-gen 1546 . . . 4 x((x = y φ) → φ)
4 19.23t 1800 . . . 4 (Ⅎxφ → (x((x = y φ) → φ) ↔ (x(x = y φ) → φ)))
53, 4mpbii 202 . . 3 (Ⅎxφ → (x(x = y φ) → φ))
61, 5syl5 28 . 2 (Ⅎxφ → ([y / x]φφ))
7 nfr 1761 . . 3 (Ⅎxφ → (φxφ))
8 stdpc4 2024 . . 3 (xφ → [y / x]φ)
97, 8syl6 29 . 2 (Ⅎxφ → (φ → [y / x]φ))
106, 9impbid 183 1 (Ⅎxφ → ([y / x]φφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  [wsb 1648 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  df-sb 1649 This theorem is referenced by:  sbf  2026  sbctt  3108
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