Theorem 3bitr3g 427

Description: More general version of 3bitr3 156. Useful for converting definitions in a formula.

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.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2		3bitr3g.2	. . 3 ⊢ (ψ ↔ θ)
3	1, 2	syl5bbr 412	. 2 ⊢ (φ → (θ ↔ χ))
4		3bitr3g.3	. 2 ⊢ (χ ↔ τ)
5	3, 4	syl6bb 414	1 ⊢ (φ → (θ ↔ τ))

Syntax hints:   → wi 2   ↔ wb 127

This theorem is referenced by:  unineq 1680  elsncg 1825  iununi 2037  erth 3219  ereldm 3222  cardeq0 3639  axpownd 3747  suplem2pr 3956  lt2sq 4414  mdsym 5784

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org