Theorem orbi1d 683

Description: Deduction adding a right 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
orbi1d	⊢ (φ → ((ψ ∨ θ) ↔ (χ ∨ θ)))

Proof of Theorem orbi1d

Step	Hyp	Ref	Expression
1		bid.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	orbi2d 682	. 2 ⊢ (φ → ((θ ∨ ψ) ↔ (θ ∨ χ)))
3		orcom 376	. 2 ⊢ ((ψ ∨ θ) ↔ (θ ∨ ψ))
4		orcom 376	. 2 ⊢ ((χ ∨ θ) ↔ (θ ∨ χ))
5	2, 3, 4	3bitr4g 279	1 ⊢ (φ → ((ψ ∨ θ) ↔ (χ ∨ θ)))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357

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-or 359

This theorem is referenced by:  orbi1  686  orbi12d  690  eueq2  3011  uneq1  3412  r19.45zv  3648  rexprg  3777  rextpg  3779  lefinlteq  4464