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Theorem sbhypf 2904
 Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3171. (Contributed by Raph Levien, 10-Apr-2004.)
Hypotheses
Ref Expression
sbhypf.1 xψ
sbhypf.2 (x = A → (φψ))
Assertion
Ref Expression
sbhypf (y = A → ([y / x]φψ))
Distinct variable groups:   x,A   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)   A(y)

Proof of Theorem sbhypf
StepHypRef Expression
1 vex 2862 . . 3 y V
2 eqeq1 2359 . . 3 (x = y → (x = Ay = A))
31, 2ceqsexv 2894 . 2 (x(x = y x = A) ↔ y = A)
4 nfs1v 2106 . . . 4 x[y / x]φ
5 sbhypf.1 . . . 4 xψ
64, 5nfbi 1834 . . 3 x([y / x]φψ)
7 sbequ12 1919 . . . . 5 (x = y → (φ ↔ [y / x]φ))
87bicomd 192 . . . 4 (x = y → ([y / x]φφ))
9 sbhypf.2 . . . 4 (x = A → (φψ))
108, 9sylan9bb 680 . . 3 ((x = y x = A) → ([y / x]φψ))
116, 10exlimi 1803 . 2 (x(x = y x = A) → ([y / x]φψ))
123, 11sylbir 204 1 (y = A → ([y / x]φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  [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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  mob2  3016  ralxpf  4827
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