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Theorem sbhypf 3550
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3908. (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 2822 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
21equsexvw 2002 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
3 nfs1v 2264 . . . 4 𝑥[𝑦 / 𝑥]𝜑
4 sbhypf.1 . . . 4 𝑥𝜓
53, 4nfbi 1895 . . 3 𝑥([𝑦 / 𝑥]𝜑𝜓)
6 sbequ12 2243 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
76bicomd 224 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
8 sbhypf.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
97, 8sylan9bb 510 . . 3 ((𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
105, 9exlimi 2207 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
112, 10sylbir 236 1 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wnf 1775  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-10 2136  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-cleq 2811
This theorem is referenced by:  mob2  3703  reu2eqd  3724  cbvrabcsfw  3921  cbvopab1  5130  ralxpf  5710  cbviotaw  6314  cbvriotaw  7112  tfisi  7562  ac6sf  9899  nn0ind-raph  12070  ac6sf2  30298  nn0min  30463  ac6gf  34888  fdc1  34902
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