orbi2d - New Foundations Explorer

	New Foundations Explorer		< Previous Next > Nearby theorems
Mirrors > Home > NFE Home > Th. List > orbi2d		GIF version

Theorem orbi2d 682

Description: Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

**Hypothesis**
Ref	Expression
bid.1	⊢ (φ → (ψ ↔ χ))

**Assertion**
Ref	Expression
orbi2d	⊢ (φ → ((θ ∨ ψ) ↔ (θ ∨ χ)))

**Proof of Theorem orbi2d**
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