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  4280  dfnfc2  4892  ndmima  6111  unfi  9214  axcclem  10562  cshw1  15001  fsumcom2  15969  fprodcom2  16180  pmtr3ncomlem1  19717  rspprop  21554  mdetunilem7  22963  cmpcov2  23738  hausflf2  24347  conway  28195  umgredg  29746  vtxdginducedm1  30154  2pthfrgrrn  30913  eqdif  33187  padct  33381  cusgredgex2  36064  f1omptsnlem  38451  igenval2  39192  mpobi123f  39286  dmqsblocks  40091  brtrclfv2  44919  clsk1indlem3  45235  permaxpow  46194  permaxpr  46195  or2expropbilem1  48307  grtriproplem  49242  mo0sn  50132