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  4283  dfnfc2  4880  ndmima  6054  unfi  9085  axcclem  10351  cshw1  14728  fsumcom2  15681  fprodcom2  15891  pmtr3ncomlem1  19352  mdetunilem7  22503  cmpcov2  23275  hausflf2  23883  conway  27711  umgredg  29087  vtxdginducedm1  29493  2pthfrgrrn  30230  eqdif  32468  cusgredgex2  35116  f1omptsnlem  37330  igenval2  38066  mpobi123f  38162  dmqsblocks  38851  brtrclfv2  43720  clsk1indlem3  44036  permaxpow  45003  permaxpr  45004  or2expropbilem1  47036  grtriproplem  47943  mo0sn  48820