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Mirrors > Home > MPE Home > Th. List > sbimd | Structured version Visualization version GIF version |
Description: Deduction substituting both sides of an implication. (Contributed by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2071. (Revised by Steven Nguyen, 9-Jul-2023.) |
Ref | Expression |
---|---|
sbimd.1 | ⊢ Ⅎ𝑥𝜑 |
sbimd.2 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
sbimd | ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbimd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | sbimd.2 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 1, 2 | alrimi 2211 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
4 | spsbim 2078 | . 2 ⊢ (∀𝑥(𝜓 → 𝜒) → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 Ⅎwnf 1791 [wsb 2070 |
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 2016 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-ex 1788 df-nf 1792 df-sb 2071 |
This theorem is referenced by: (None) |
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