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Theorem sbhypf 3543
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3926. (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 2073 . 2 ([𝑦 / 𝑥]𝑥 = 𝐴 → [𝑦 / 𝑥](𝜑𝜓))
3 eqsb1 2866 . 2 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
4 sbhypf.1 . . . 4 𝑥𝜓
54sbf 2270 . . 3 ([𝑦 / 𝑥]𝜓𝜓)
65sblbis 2308 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))
72, 3, 63imtr3i 291 1 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wnf 1782  [wsb 2063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-sb 2064  df-cleq 2728
This theorem is referenced by:  mob2  3720  reu2eqd  3741  cbvrabcsfw  3939  cbvopab1  5216  ralxpf  5856  cbviotaw  6520  cbvriotaw  7398  tfisi  7881  ac6sf  10530  nn0ind-raph  12720  ac6sf2  32635  nn0min  32823  ac6gf  37740  fdc1  37754
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