Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbi2ALT | Structured version Visualization version GIF version |
Description: Alternate version of sbi2 2309. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1.p5 | ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
dfsb1.s2 | ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) |
dfsb1.im | ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓)))) |
Ref | Expression |
---|---|
sbi2ALT | ⊢ ((𝜃 → 𝜏) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsb1.p5 | . . . 4 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | biid 263 | . . . 4 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑))) | |
3 | 1, 2 | sbnALT 2594 | . . 3 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ¬ 𝜃) |
4 | dfsb1.im | . . . 4 ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓)))) | |
5 | pm2.21 123 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
6 | 2, 4, 5 | sbimiALT 2576 | . . 3 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) → 𝜂) |
7 | 3, 6 | sylbir 237 | . 2 ⊢ (¬ 𝜃 → 𝜂) |
8 | dfsb1.s2 | . . 3 ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) | |
9 | ax-1 6 | . . 3 ⊢ (𝜓 → (𝜑 → 𝜓)) | |
10 | 8, 4, 9 | sbimiALT 2576 | . 2 ⊢ (𝜏 → 𝜂) |
11 | 7, 10 | ja 188 | 1 ⊢ ((𝜃 → 𝜏) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∃wex 1779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 |
This theorem is referenced by: sbimALT 2607 |
Copyright terms: Public domain | W3C validator |