Theorem sylbbr 239

Description: A mixed syllogism inference from two biconditionals.

Note on the various syllogism-like statements in set.mm. The hypothetical syllogism syl 18 infers an implication from two implications (and there are 3syl 19 and 4syl 20 for chaining more inferences). There are four inferences inferring an implication from one implication and one biconditional: sylbi 220, sylib 221, sylbir 238, sylibr 237; four inferences inferring an implication from two biconditionals: sylbb 222, sylbbr 239, sylbb1 240, sylbb2 241; four inferences inferring a biconditional from two biconditionals: bitri 278, bitr2i 279, bitr3i 280, bitr4i 281 (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld 48, syl5 35, syl6 36, mpbid 235, bitrd 282, bitrid 286, bitrdi 290 and variants. (Contributed by BJ, 21-Apr-2019.)

Hypotheses

Ref	Expression
sylbbr.1	⊢ (𝜑 ↔ 𝜓)
sylbbr.2	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
sylbbr	⊢ (𝜒 → 𝜑)

Proof of Theorem sylbbr

Step	Hyp	Ref	Expression
1		sylbbr.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
2	1	biimpri 231	. 2 ⊢ (𝜒 → 𝜓)
3		sylbbr.1	. 2 ⊢ (𝜑 ↔ 𝜓)
4	2, 3	sylibr 237	1 ⊢ (𝜒 → 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  bitri  278  euelss  4277  dfnfc2  4888  ndmima  6093  unfi  9164  axcclem  10506  cshw1  14940  fsumcom2  15907  fprodcom2  16118  pmtr3ncomlem1  19648  rspprop  21485  mdetunilem7  22894  cmpcov2  23669  hausflf2  24278  conway  28098  umgredg  29649  vtxdginducedm1  30057  2pthfrgrrn  30816  eqdif  33048  padct  33243  cusgredgex2  35828  f1omptsnlem  38179  igenval2  38920  mpobi123f  39014  dmqsblocks  39819  brtrclfv2  44671  clsk1indlem3  44987  permaxpow  45936  permaxpr  45937  or2expropbilem1  48024  grtriproplem  48959  mo0sn  49848