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Mirrors > Home > MPE Home > Th. List > sbiedwOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbiedw 2332 as of 28-Jan-2024. (Contributed by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbiedw.1 | ⊢ Ⅎ𝑥𝜑 |
sbiedw.2 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
sbiedw.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
sbiedwOLD | ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbiedw.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | sbrim 2313 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) |
3 | sbiedw.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
4 | 1, 3 | nfim1 2199 | . . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜒) |
5 | sbiedw.3 | . . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
6 | 5 | com12 32 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒))) |
7 | 6 | pm5.74d 275 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
8 | 4, 7 | sbiev 2330 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) |
9 | 2, 8 | bitr3i 279 | . 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → 𝜒)) |
10 | 9 | pm5.74ri 274 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 Ⅎwnf 1784 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: (None) |
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