Theorem 3bitr3g 222

Description: More general version of 3bitr3i 210. 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 194	. 2 ⊢ (𝜑 → (𝜃 ↔ 𝜒))
4		3bitr3g.3	. 2 ⊢ (𝜒 ↔ 𝜏)
5	3, 4	bitrdi 196	1 ⊢ (𝜑 → (𝜃 ↔ 𝜏))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  con2bidc  887  sbal1yz  2061  sbal1  2062  dfsbcq2  3054  iindif2m  4075  opeqex  4385  rabxfrd  4610  eqbrrdv  4867  eqbrrdiv  4868  opelco2g  4943  opelcnvg  4955  ralrnmpt  5841  rexrnmpt  5842  fliftcnv  5991  eusvobj2  6061  f1od2  6461  ottposg  6516  ercnv  6818  exmidpw  7205  djuf1olem  7383  fzen  10426  fihasheq0  11210  divalgb  12670  isprm3  12874  eldvap  15706