Theorem sylbb1 236

Description: A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)

Hypotheses

Ref	Expression
sylbb1.1	⊢ (𝜑 ↔ 𝜓)
sylbb1.2	⊢ (𝜑 ↔ 𝜒)

Assertion

Ref	Expression
sylbb1	⊢ (𝜓 → 𝜒)

Proof of Theorem sylbb1

Step	Hyp	Ref	Expression
1		sylbb1.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	biimpri 227	. 2 ⊢ (𝜓 → 𝜑)
3		sylbb1.2	. 2 ⊢ (𝜑 ↔ 𝜒)
4	2, 3	sylib 217	1 ⊢ (𝜓 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 205

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 206

This theorem is referenced by:  fsuppmapnn0fiubex  13957  rrxcph  24909  volun  25062  umgrislfupgr  28383  usgrislfuspgr  28444  wlkp1lem8  28937  elwwlks2s3  29205  eupthp1  29469  cnvbraval  31363  ballotlemfp1  33490  finixpnum  36473  fin2so  36475  matunitlindflem1  36484  oeord2com  42061  clsf2  42877  ellimcabssub0  44333  sge0iunmpt  45134  icceuelpartlem  46103  nnsum4primesodd  46464  nnsum4primesoddALTV  46465