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Theorem sbhypf 3441
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3747. (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 eqeq1 2803 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
21equsexvw 2104 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
3 nfs1v 2303 . . . 4 𝑥[𝑦 / 𝑥]𝜑
4 sbhypf.1 . . . 4 𝑥𝜓
53, 4nfbi 2003 . . 3 𝑥([𝑦 / 𝑥]𝜑𝜓)
6 sbequ12 2278 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
76bicomd 215 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
8 sbhypf.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
97, 8sylan9bb 506 . . 3 ((𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
105, 9exlimi 2252 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
112, 10sylbir 227 1 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wex 1875  wnf 1879  [wsb 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-cleq 2792
This theorem is referenced by:  mob2  3582  reu2eqd  3601  cbvmptf  4941  ralxpf  5472  tfisi  7292  ac6sf  9599  nn0ind-raph  11767  ac6sf2  29948  nn0min  30085  ac6gf  34015  fdc1  34029
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