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  4277  dfnfc2  4888  ndmima  6098  unfi  9186  axcclem  10535  cshw1  14973  fsumcom2  15940  fprodcom2  16151  pmtr3ncomlem1  19687  rspprop  21524  mdetunilem7  22933  cmpcov2  23708  hausflf2  24317  conway  28165  umgredg  29716  vtxdginducedm1  30124  2pthfrgrrn  30883  eqdif  33157  padct  33352  cusgredgex2  35949  f1omptsnlem  38302  igenval2  39043  mpobi123f  39137  dmqsblocks  39942  brtrclfv2  44770  clsk1indlem3  45086  permaxpow  46042  permaxpr  46043  or2expropbilem1  48145  grtriproplem  49080  mo0sn  49969