Theorem 3bitr3g 278

Description: More general version of 3bitr3i 266. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)

Hypotheses

Ref	Expression
3bitr3g.1	⊢ (φ → (ψ ↔ χ))
3bitr3g.2	⊢ (ψ ↔ θ)
3bitr3g.3	⊢ (χ ↔ τ)

Assertion

Ref	Expression
3bitr3g	⊢ (φ → (θ ↔ τ))

Proof of Theorem 3bitr3g

Step	Hyp	Ref	Expression
1		3bitr3g.2	. . 3 ⊢ (ψ ↔ θ)
2		3bitr3g.1	. . 3 ⊢ (φ → (ψ ↔ χ))
3	1, 2	syl5bbr 250	. 2 ⊢ (φ → (θ ↔ χ))
4		3bitr3g.3	. 2 ⊢ (χ ↔ τ)
5	3, 4	syl6bb 252	1 ⊢ (φ → (θ ↔ τ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  notbid  285  cador  1391  equequ2  1686  dfsbcq2  3050  unineq  3506  iindif2  4036  isoini  5498  brcupg  5815  enprmaplem3  6079  nmembers1lem3  6271