Theorem syl7bi 255

Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993.)

Hypotheses

Ref	Expression
syl7bi.1	⊢ (𝜑 ↔ 𝜓)
syl7bi.2	⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏)))

Assertion

Ref	Expression
syl7bi	⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏)))

Proof of Theorem syl7bi

Step	Hyp	Ref	Expression
1		syl7bi.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	biimpi 215	. 2 ⊢ (𝜑 → 𝜓)
3		syl7bi.2	. 2 ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏)))
4	2, 3	syl7 74	1 ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏)))

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:  3jao  1425  rspct  3552  zfpair  5353  gruen  10618  axpre-sup  10975  nn0lt2  12433  fzofzim  13484  ndvdssub  16167  cyccom  18871  alexsubALT  23251  clwlkclwwlklem2a  28411  erclwwlktr  28435  erclwwlkntr  28484  fmlasuc  33397  dfon2lem8  33815  prtlem15  37089  prtlem18  37091  2reuimp0  44850