Theorem imbi2i 226

Description: Introduce an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)

Hypothesis

Ref	Expression
bi.a	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
imbi2i	⊢ ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓))

Proof of Theorem imbi2i

Step	Hyp	Ref	Expression
1		bi.a	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	a1i 9	. 2 ⊢ (𝜒 → (𝜑 ↔ 𝜓))
3	2	pm5.74i 180	1 ⊢ ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  imbi12i  239  anidmdbi  402  nan  703  sbcof2  1863  sblimv  1950  sbhb  2000  sblim  2017  2sb6  2044  sbcom2v  2045  2sb6rf  2050  eu1  2111  moabs  2136  mo3h  2140  moanim  2161  2moswapdc  2177  r2alf  2567  r19.21t  2625  rspc2gv  2942  reu2  3014  reu8  3022  2reuswapdc  3030  2rmorex  3032  dfdif3  3339  ssconb  3362  ssin  3453  reldisj  3575  ssundifim  3608  ralm  3628  unissb  3960  repizf2lem  4293  elirr  4683  en2lp  4696  tfi  4724  ssrel  4858  ssrel2  4860  fncnv  5442  fun11  5443  axcaucvglemres  8256  axpre-suploc  8259  suprzclex  9723  raluz2  9958  supinfneg  9974  infsupneg  9975  infssuzex  10644  bezoutlemmain  12753  isprm2  12873  lmres  15272  ivthdich  15677  limcdifap  15686