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  880  sbal1yz  2052  sbal1  2053  dfsbcq2  3032  iindif2m  4036  opeqex  4340  rabxfrd  4564  eqbrrdv  4821  eqbrrdiv  4822  opelco2g  4896  opelcnvg  4908  ralrnmpt  5785  rexrnmpt  5786  fliftcnv  5931  eusvobj2  5999  f1od2  6395  ottposg  6416  ercnv  6718  exmidpw  7093  djuf1olem  7243  fzen  10268  fihasheq0  11045  divalgb  12476  isprm3  12680  eldvap  15396