Theorem syl5ibr 145

Description: A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (Revised by NM, 22-Sep-2013.)

Hypotheses

Ref	Expression
syl5ibr.1	⊢ (φ → θ)
syl5ibr.2	⊢ (χ → (ψ ↔ θ))

Assertion

Ref	Expression
syl5ibr	⊢ (χ → (φ → ψ))

Proof of Theorem syl5ibr

Step	Hyp	Ref	Expression
1		syl5ibr.1	. 2 ⊢ (φ → θ)
2		syl5ibr.2	. . 3 ⊢ (χ → (ψ ↔ θ))
3	2	bicomd 129	. 2 ⊢ (χ → (θ ↔ ψ))
4	1, 3	syl5ib 143	1 ⊢ (χ → (φ → ψ))

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  syl5ibrcom  146  biimprd  147  nbn2  612  limelon  4102  eldifpw  4174  ssonuni  4180  peano2  4261  limom  4279  elrnmpt1  4528  cnveqb  4719  cnveq0  4720  relcoi1  4792  ndmfvg  5147  ffvresb  5271  caovord3d  5613  poxp  5794  nnm0r  5997  nnacl  5998  nnacom  6002  nnaass  6003  nndi  6004  nnmass  6005  nnmsucr  6006  nnmcom  6007  brecop  6132  ecopovtrn  6139  ecopovtrng  6142  fundmen  6222  mulcmpblnq  6352  ordpipqqs  6358  mulcmpblnq0  6426  genpprecll  6496  genppreclu  6497  addcmpblnr  6647  ax1rid  6741  axpre-mulgt0  6751  cnegexlem1  6963  msqge0  7380  mulge0  7383  ltleap  7393  nnmulcl  7696  nnsub  7713  elnn0z  8014  ztri3or0  8043  nneoor  8096  uz11  8251  xltnegi  8498  frec2uzuzd  8849  iseqfveq2  8885  m1expcl2  8911  expadd  8931  expmul  8934  bj-om  9371  peano5set  9374  bj-inf2vnlem2  9401  bj-inf2vn  9404  bj-inf2vn2  9405