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  4285  dfnfc2  4886  ndmima  6063  unfi  9099  axcclem  10371  cshw1  14749  fsumcom2  15701  fprodcom2  15911  pmtr3ncomlem1  19406  mdetunilem7  22566  cmpcov2  23338  hausflf2  23946  conway  27779  umgredg  29215  vtxdginducedm1  29621  2pthfrgrrn  30361  eqdif  32597  cusgredgex2  35319  f1omptsnlem  37543  igenval2  38269  mpobi123f  38365  dmqsblocks  39170  brtrclfv2  44035  clsk1indlem3  44351  permaxpow  45317  permaxpr  45318  or2expropbilem1  47345  grtriproplem  48252  mo0sn  49128