Theorem orbi1i 679

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

Hypothesis

Ref	Expression
orbi2i.1	⊢ (φ ↔ ψ)

Assertion

Ref	Expression
orbi1i	⊢ ((φ ∨ χ) ↔ (ψ ∨ χ))

Proof of Theorem orbi1i

Step	Hyp	Ref	Expression
1		orcom 646	. 2 ⊢ ((φ ∨ χ) ↔ (χ ∨ φ))
2		orbi2i.1	. . 3 ⊢ (φ ↔ ψ)
3	2	orbi2i 678	. 2 ⊢ ((χ ∨ φ) ↔ (χ ∨ ψ))
4		orcom 646	. 2 ⊢ ((χ ∨ ψ) ↔ (ψ ∨ χ))
5	1, 3, 4	3bitri 195	1 ⊢ ((φ ∨ χ) ↔ (ψ ∨ χ))

Syntax hints:   ↔ wb 98   ∨ wo 628

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  orbi12i  680  orordi  689  dcan  841  3or6  1217  19.45  1570  sbequilem  1716  unass  3094  frecsuc  5930  elznn0nn  8035