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  4262  dfnfc2  4862  ndmima  6057  unfi  9094  axcclem  10368  cshw1  14773  fsumcom2  15725  fprodcom2  15938  pmtr3ncomlem1  19437  mdetunilem7  22571  cmpcov2  23343  hausflf2  23951  conway  27759  umgredg  29195  vtxdginducedm1  29600  2pthfrgrrn  30340  eqdif  32577  padct  32779  cusgredgex2  35293  f1omptsnlem  37640  igenval2  38375  mpobi123f  38471  dmqsblocks  39276  brtrclfv2  44142  clsk1indlem3  44458  permaxpow  45424  permaxpr  45425  or2expropbilem1  47468  grtriproplem  48403  mo0sn  49279