Theorem mpbii 148

Description: An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)

Hypotheses

Ref	Expression
mpbii.min	⊢ 𝜓
mpbii.maj	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
mpbii	⊢ (𝜑 → 𝜒)

Proof of Theorem mpbii

Step	Hyp	Ref	Expression
1		mpbii.min	. . 3 ⊢ 𝜓
2	1	a1i 9	. 2 ⊢ (𝜑 → 𝜓)
3		mpbii.maj	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	mpbid 147	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm2.26dc  912  orandc  945  19.9ht  1687  ax11v2  1866  ax11v  1873  ax11ev  1874  equs5or  1876  nfsbxy  1993  nfsbxyt  1994  nfabdw  2391  eqvisset  2810  vtoclgf  2859  vtoclg1f  2860  eueq3dc  2977  mo2icl  2982  csbiegf  3168  un00  3538  sneqr  3838  preqr1  3846  preq12b  3848  prel12  3849  nfopd  3874  ssex  4221  exmidundif  4290  iunpw  4571  nfimad  5077  dfrel2  5179  funsng  5367  cnvresid  5395  nffvd  5641  fnbrfvb  5674  funfvop  5749  acexmidlema  5998  tposf12  6421  supsnti  7180  pr2cv1  7376  exmidonfinlem  7379  sucpw1ne3  7425  sucpw1nel3  7426  recidnq  7588  ltaddnq  7602  ltadd1sr  7971  suplocsrlempr  8002  pncan3  8362  divcanap2  8835  ltp1  8999  ltm1  9001  recreclt  9055  nn0ind-raph  9572  2tnp1ge0ge0  10529  iswrdiz  11086  fsumcnv  11956  fprodcnv  12144  ef01bndlem  12275  sin01gt0  12281  cos01gt0  12282  ltoddhalfle  12412  bezoutlemnewy  12525  isprm5  12672  4sqlem12  12933  gsumval2  13438  nmznsg  13758  tangtx  15520  gausslemma2dlem1a  15745  lgseisenlem4  15760  2lgslem3a  15780  2lgslem3b  15781  2lgslem3c  15782  2lgslem3d  15783  bdsepnft  16274  bdssex  16289  bj-inex  16294  bj-d0clsepcl  16312  bj-2inf  16325  bj-inf2vnlem2  16358