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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. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.) Revise df-sb 2016. (Revised by Wolf Lammen, 28-Jul-2023.) |
Ref | Expression |
---|---|
sbequ8 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsb1 2451 | . . 3 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | |
2 | 1 | a1bi 355 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑)) |
3 | sbim 2237 | . 2 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑) ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑)) | |
4 | 2, 3 | bitr4i 270 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 [wsb 2015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-10 2079 ax-12 2106 ax-13 2301 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-ex 1743 df-nf 1747 df-sb 2016 |
This theorem is referenced by: (None) |
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