Theorem bibi1d 310

Description: Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
imbid.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
bibi1d	⊢ (φ → ((ψ ↔ θ) ↔ (χ ↔ θ)))

Proof of Theorem bibi1d

Step	Hyp	Ref	Expression
1		imbid.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	bibi2d 309	. 2 ⊢ (φ → ((θ ↔ ψ) ↔ (θ ↔ χ)))
3		bicom 191	. 2 ⊢ ((ψ ↔ θ) ↔ (θ ↔ ψ))
4		bicom 191	. 2 ⊢ ((χ ↔ θ) ↔ (θ ↔ χ))
5	2, 3, 4	3bitr4g 279	1 ⊢ (φ → ((ψ ↔ θ) ↔ (χ ↔ θ)))

Syntax hints:   → wi 4   ↔ wb 176

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

This theorem is referenced by:  bibi12d  312  bibi1  317  biass  348  eubid  2211  axext3  2336  bm1.1  2338  eqeq1  2359  pm13.183  2980  elabgt  2983  mob  3019  sbctt  3109  sbcabel  3124  br1stg  4731  isoeq2  5484  extd  5924