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| 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.) | 
| Ref | Expression | 
|---|---|
| sbhypf.1 | ⊢ Ⅎ𝑥𝜓 | 
| sbhypf.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| sbhypf | ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbhypf.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 2 | 1 | sbimi 2073 | . 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 → [𝑦 / 𝑥](𝜑 ↔ 𝜓)) | 
| 3 | eqsb1 2866 | . 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴) | |
| 4 | sbhypf.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 5 | 4 | sbf 2270 | . . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓) | 
| 6 | 5 | sblbis 2308 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) | 
| 7 | 2, 3, 6 | 3imtr3i 291 | 1 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 Ⅎwnf 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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