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)

Syntax hints:   → wi 4   ↔ wb 209

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 210

This theorem is referenced by:  bitri  278  euelss  4284  dfnfc2  4893  ndmima  6105  unfi  9154  axcclem  10440  cshw1  14858  fsumcom2  15824  fprodcom2  16037  pmtr3ncomlem1  19542  rspprop  21349  mdetunilem7  22754  cmpcov2  23526  hausflf2  24134  conway  27948  umgredg  29454  vtxdginducedm1  29859  2pthfrgrrn  30599  eqdif  32831  padct  33029  cusgredgex2  35569  f1omptsnlem  37926  igenval2  38661  mpobi123f  38757  dmqsblocks  39562  brtrclfv2  44401  clsk1indlem3  44717  permaxpow  45666  permaxpr  45667  or2expropbilem1  47714  grtriproplem  48649  mo0sn  49539