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)))

Syntax hints:  ¬ wn 3   → 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:  orbi1d  683  orbi12d  690  cad1  1398  eueq2  3010  sbc2or  3054  rexprg  3776  rextpg  3778  clos1basesucg  5884  nc0suc  6217  nmembers1lem3  6270