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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 2139. (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 2305 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbrim.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | sbf 2264 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
4 | 3 | imbi1i 352 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
5 | 1, 4 | bitri 277 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 Ⅎwnf 1778 [wsb 2063 |
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 1905 ax-6 1964 ax-7 2009 ax-10 2139 ax-12 2170 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1775 df-nf 1779 df-sb 2064 |
This theorem is referenced by: sbiedwOLD 2327 sbied 2539 sbco2d 2548 2mos 2728 |
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