Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbimALT | Structured version Visualization version GIF version |
Description: Alternate version of sbim 2310. (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 |
---|---|
sbimALT | ⊢ (𝜂 ↔ (𝜃 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsb1.p5 | . . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | dfsb1.s2 | . . 3 ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) | |
3 | dfsb1.im | . . 3 ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → 𝜓)))) | |
4 | 1, 2, 3 | sbi1ALT 2605 | . 2 ⊢ (𝜂 → (𝜃 → 𝜏)) |
5 | 1, 2, 3 | sbi2ALT 2606 | . 2 ⊢ ((𝜃 → 𝜏) → 𝜂) |
6 | 4, 5 | impbii 211 | 1 ⊢ (𝜂 ↔ (𝜃 → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → 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: sbrimALT 2608 sbanALT 2609 sbbiALT 2610 |
Copyright terms: Public domain | W3C validator |