![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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. (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 2238 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbrim.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | sbf 2200 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
4 | 3 | imbi1i 342 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
5 | 1, 4 | bitri 267 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 Ⅎwnf 1747 [wsb 2016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-10 2080 ax-12 2107 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-ex 1744 df-nf 1748 df-sb 2017 |
This theorem is referenced by: sbied 2470 sbco2d 2479 2mos 2680 |
Copyright terms: Public domain | W3C validator |