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Mirrors > Home > MPE Home > Th. List > sbrim | Structured version Visualization version GIF version |
Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. See sbrimv 2308 for a version with disjoint variables not requiring ax-10 2143. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Ref | Expression |
---|---|
sbrim.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
sbrim | ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbim 2306 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbrim.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | sbf 2269 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
4 | 3 | imbi1i 353 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
5 | 1, 4 | bitri 278 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1791 [wsb 2072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-10 2143 ax-12 2177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-sb 2073 |
This theorem is referenced by: sbiedwOLD 2317 sbied 2506 sbco2d 2515 2mos 2650 |
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