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Mirrors > Home > MPE Home > Th. List > sbequ8 | Structured version Visualization version GIF version |
Description: Elimination of equality from antecedent after substitution. Usage of this theorem is discouraged because it depends on ax-13 2367. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.) Revise df-sb 2061. (Revised by Wolf Lammen, 28-Jul-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbequ8 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsb1 2486 | . . 3 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | |
2 | 1 | a1bi 362 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑)) |
3 | sbim 2293 | . 2 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑) ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑)) | |
4 | 2, 3 | bitr4i 278 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: (None) |
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