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Mirrors > Home > MPE Home > Th. List > nanbi2 | Structured version Visualization version GIF version |
Description: Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) |
Ref | Expression |
---|---|
nanbi2 | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nanbi1 1447 | . 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒))) | |
2 | nancom 1442 | . 2 ⊢ ((𝜒 ⊼ 𝜑) ↔ (𝜑 ⊼ 𝜒)) | |
3 | nancom 1442 | . 2 ⊢ ((𝜒 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜒)) | |
4 | 1, 2, 3 | 3bitr4g 302 | 1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ⊼ wnan 1439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-an 385 df-nan 1440 |
This theorem is referenced by: nanbi12 1449 nanbi2i 1451 nanbi2d 1454 nabi2 31560 rp-fakenanass 36879 |
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