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Mirrors > Home > NFE Home > Th. List > orbi2d | GIF version |
Description: Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
bid.1 | ⊢ (φ → (ψ ↔ χ)) |
Ref | Expression |
---|---|
orbi2d | ⊢ (φ → ((θ ∨ ψ) ↔ (θ ∨ χ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bid.1 | . . 3 ⊢ (φ → (ψ ↔ χ)) | |
2 | 1 | imbi2d 307 | . 2 ⊢ (φ → ((¬ θ → ψ) ↔ (¬ θ → χ))) |
3 | df-or 359 | . 2 ⊢ ((θ ∨ ψ) ↔ (¬ θ → ψ)) | |
4 | df-or 359 | . 2 ⊢ ((θ ∨ χ) ↔ (¬ θ → χ)) | |
5 | 2, 3, 4 | 3bitr4g 279 | 1 ⊢ (φ → ((θ ∨ ψ) ↔ (θ ∨ χ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∨ wo 357 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 |
This theorem depends on definitions: df-bi 177 df-or 359 |
This theorem is referenced by: orbi1d 683 orbi12d 690 cad1 1398 eueq2 3010 sbc2or 3054 rexprg 3776 rextpg 3778 clos1basesucg 5884 nc0suc 6217 nmembers1lem3 6270 |
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