Theorem sylbbr 236

Description: A mixed syllogism inference from two biconditionals.

Note on the various syllogism-like statements in set.mm. The hypothetical syllogism syl 17 infers an implication from two implications (and there are 3syl 18 and 4syl 19 for chaining more inferences). There are four inferences inferring an implication from one implication and one biconditional: sylbi 217, sylib 218, sylbir 235, sylibr 234; four inferences inferring an implication from two biconditionals: sylbb 219, sylbbr 236, sylbb1 237, sylbb2 238; four inferences inferring a biconditional from two biconditionals: bitri 275, bitr2i 276, bitr3i 277, bitr4i 278 (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld 47, syl5 34, syl6 35, mpbid 232, bitrd 279, bitrid 283, bitrdi 287 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 228	. 2 ⊢ (𝜒 → 𝜓)
3		sylbbr.1	. 2 ⊢ (𝜑 ↔ 𝜓)
4	2, 3	sylibr 234	1 ⊢ (𝜒 → 𝜑)

Syntax hints:   → wi 4   ↔ wb 206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207

This theorem is referenced by:  bitri  275  euelss  4332  dfnfc2  4929  ndmima  6121  unfi  9211  axcclem  10497  cshw1  14860  fsumcom2  15810  fprodcom2  16020  pmtr3ncomlem1  19491  mdetunilem7  22624  cmpcov2  23398  hausflf2  24006  conway  27844  umgredg  29155  vtxdginducedm1  29561  2pthfrgrrn  30301  eqdif  32538  cusgredgex2  35128  f1omptsnlem  37337  igenval2  38073  mpobi123f  38169  brtrclfv2  43740  clsk1indlem3  44056  or2expropbilem1  47044  grtriproplem  47906  mo0sn  48735