Theorem 3bitr3d 423

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula.

Hypotheses

Ref	Expression
3bitr3d.1	⊢ (φ → (ψ ↔ χ))
3bitr3d.2	⊢ (φ → (ψ ↔ θ))
3bitr3d.3	⊢ (φ → (χ ↔ τ))

Assertion

Ref	Expression
3bitr3d	⊢ (φ → (θ ↔ τ))

Proof of Theorem 3bitr3d

Step	Hyp	Ref	Expression
1		3bitr3d.2	. . 3 ⊢ (φ → (ψ ↔ θ))
2		3bitr3d.1	. . 3 ⊢ (φ → (ψ ↔ χ))
3	1, 2	bitr3d 408	. 2 ⊢ (φ → (θ ↔ χ))
4		3bitr3d.3	. 2 ⊢ (φ → (χ ↔ τ))
5	3, 4	bitrd 406	1 ⊢ (φ → (θ ↔ τ))

Syntax hints:   → wi 2   ↔ wb 127

This theorem is referenced by:  ordsucun 2333  cbvfo 2923  eloprabg 3035  prlem936a 3947  ltaddsubt 4357  rebtwnz 4620  hi2eqt 5059

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

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