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Mirrors > Home > NFE Home > Th. List > nanbi12i | GIF version |
Description: Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.) |
Ref | Expression |
---|---|
nanbii.1 | ⊢ (φ ↔ ψ) |
nanbi12i.2 | ⊢ (χ ↔ θ) |
Ref | Expression |
---|---|
nanbi12i | ⊢ ((φ ⊼ χ) ↔ (ψ ⊼ θ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nanbii.1 | . 2 ⊢ (φ ↔ ψ) | |
2 | nanbi12i.2 | . 2 ⊢ (χ ↔ θ) | |
3 | nanbi12 1297 | . 2 ⊢ (((φ ↔ ψ) ∧ (χ ↔ θ)) → ((φ ⊼ χ) ↔ (ψ ⊼ θ))) | |
4 | 1, 2, 3 | mp2an 653 | 1 ⊢ ((φ ⊼ χ) ↔ (ψ ⊼ θ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ⊼ wnan 1287 |
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 177 df-an 360 df-nan 1288 |
This theorem is referenced by: (None) |
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