Theorem 3bitr4d 276

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)

Hypotheses

Ref	Expression
3bitr4d.1	⊢ (φ → (ψ ↔ χ))
3bitr4d.2	⊢ (φ → (θ ↔ ψ))
3bitr4d.3	⊢ (φ → (τ ↔ χ))

Assertion

Ref	Expression
3bitr4d	⊢ (φ → (θ ↔ τ))

Proof of Theorem 3bitr4d

Step	Hyp	Ref	Expression
1		3bitr4d.2	. 2 ⊢ (φ → (θ ↔ ψ))
2		3bitr4d.1	. . 3 ⊢ (φ → (ψ ↔ χ))
3		3bitr4d.3	. . 3 ⊢ (φ → (τ ↔ χ))
4	2, 3	bitr4d 247	. 2 ⊢ (φ → (ψ ↔ τ))
5	1, 4	bitrd 244	1 ⊢ (φ → (θ ↔ τ))

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:  sbcom  2089  sbcom2  2114  r19.12sn  3790  lefinlteq  4464  eqtfinrelk  4487  tfinlefin  4503  opbrop  4842  fvopab3g  5387  unpreima  5409  inpreima  5410  respreima  5411  fconst5  5456  isotr  5496  ncseqnc  6129