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Theorem sbhypf 3225
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3517. (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 2613 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
21equsexvw 1918 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
3 nfs1v 2424 . . . 4 𝑥[𝑦 / 𝑥]𝜑
4 sbhypf.1 . . . 4 𝑥𝜓
53, 4nfbi 1820 . . 3 𝑥([𝑦 / 𝑥]𝜑𝜓)
6 sbequ12 2096 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
76bicomd 211 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
8 sbhypf.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
97, 8sylan9bb 731 . . 3 ((𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
105, 9exlimi 2072 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
112, 10sylbir 223 1 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wnf 1698  [wsb 1866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-cleq 2602
This theorem is referenced by:  mob2  3352  reu2eqd  3369  cbvmptf  4670  ralxpf  5178  tfisi  6927  ac6sf  9171  nn0ind-raph  11309  ac6sf2  28616  nn0min  28760  ac6gf  32493  fdc1  32508
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