Theorem bitr2d 178

Description: Deduction form of bitr2i 174. (Contributed by NM, 9-Jun-2004.)

Hypotheses

Ref	Expression
bitr2d.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
bitr2d.2	⊢ (𝜑 → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
bitr2d	⊢ (𝜑 → (𝜃 ↔ 𝜓))

Proof of Theorem bitr2d

Step	Hyp	Ref	Expression
1		bitr2d.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		bitr2d.2	. . 3 ⊢ (𝜑 → (𝜒 ↔ 𝜃))
3	1, 2	bitrd 177	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
4	3	bicomd 129	1 ⊢ (𝜑 → (𝜃 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  3bitrrd  204  3bitr2rd  206  pm5.18dc  777  drex1  1679  elrnmpt1  4585  xpopth  5802  sbcopeq1a  5813  ltnnnq  6521  ltaddsub  7431  leaddsub  7433  posdif  7450  lesub1  7451  ltsub1  7453  lesub0  7474  possumd  7560  ltdivmul  7842  ledivmul  7843  zlem1lt  8300  zltlem1  8301  fzrev2  8947  fz1sbc  8958  elfzp1b  8959  qtri3or  9098  sumsqeq0  9332  sqrtle  9634  sqrtlt  9635  absgt0ap  9695