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  4281  dfnfc2  4892  ndmima  6103  unfi  9168  axcclem  10462  cshw1  14895  fsumcom2  15862  fprodcom2  16075  pmtr3ncomlem1  19601  rspprop  21434  mdetunilem7  22841  cmpcov2  23616  hausflf2  24225  conway  28042  umgredg  29581  vtxdginducedm1  29989  2pthfrgrrn  30748  eqdif  32980  padct  33176  cusgredgex2  35708  f1omptsnlem  38077  igenval2  38803  mpobi123f  38897  dmqsblocks  39702  brtrclfv2  44554  clsk1indlem3  44870  permaxpow  45819  permaxpr  45820  or2expropbilem1  47907  grtriproplem  48842  mo0sn  49731