Theorem 3bitr3g 221

Description: More general version of 3bitr3i 209. 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	bitr3id 193	. 2 ⊢ (𝜑 → (𝜃 ↔ 𝜒))
4		3bitr3g.3	. 2 ⊢ (𝜒 ↔ 𝜏)
5	3, 4	bitrdi 195	1 ⊢ (𝜑 → (𝜃 ↔ 𝜏))

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  con2bidc  861  sbal1yz  1981  sbal1  1982  dfsbcq2  2940  iindif2m  3916  opeqex  4209  rabxfrd  4428  eqbrrdv  4682  eqbrrdiv  4683  opelco2g  4753  opelcnvg  4765  ralrnmpt  5608  rexrnmpt  5609  fliftcnv  5742  eusvobj2  5807  f1od2  6179  ottposg  6199  ercnv  6498  exmidpw  6850  djuf1olem  6991  fzen  9938  fihasheq0  10661  divalgb  11808  isprm3  11986  eldvap  13022