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Theorem sbhypf 3538
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3921. (Contributed by Raph Levien, 10-Apr-2004.) (Proof shortened by Wolf Lammen, 25-Jan-2025.)
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 sbhypf.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
21sbimi 2077 . 2 ([𝑦 / 𝑥]𝑥 = 𝐴 → [𝑦 / 𝑥](𝜑𝜓))
3 eqsb1 2859 . 2 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
4 sbhypf.1 . . . 4 𝑥𝜓
54sbf 2262 . . 3 ([𝑦 / 𝑥]𝜓𝜓)
65sblbis 2305 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))
72, 3, 63imtr3i 290 1 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wnf 1785  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068  df-cleq 2724
This theorem is referenced by:  mob2  3710  reu2eqd  3731  cbvrabcsfw  3936  cbvopab1  5222  ralxpf  5844  cbviotaw  6499  cbvriotaw  7370  tfisi  7844  ac6sf  10480  nn0ind-raph  12658  ac6sf2  31836  nn0min  32013  ac6gf  36588  fdc1  36602
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