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Mirrors > Home > MPE Home > Th. List > sbbiiALT | Structured version Visualization version GIF version |
Description: Alternate version of sbbii 2080. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1.ph | ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
dfsb1.ps | ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) |
sbbiiALT.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
sbbiiALT | ⊢ (𝜃 ↔ 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsb1.ph | . . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | dfsb1.ps | . . 3 ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) | |
3 | sbbiiALT.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
4 | 3 | biimpi 218 | . . 3 ⊢ (𝜑 → 𝜓) |
5 | 1, 2, 4 | sbimiALT 2576 | . 2 ⊢ (𝜃 → 𝜏) |
6 | 3 | biimpri 230 | . . 3 ⊢ (𝜓 → 𝜑) |
7 | 2, 1, 6 | sbimiALT 2576 | . 2 ⊢ (𝜏 → 𝜃) |
8 | 5, 7 | impbii 211 | 1 ⊢ (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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 |
This theorem is referenced by: sbanALT 2609 sbbiALT 2610 sb7fALT 2615 |
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