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 216	. 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 206

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 207

This theorem is referenced by:  3jao  1427  rspct  3577  zfpair  5379  gruen  10772  axpre-sup  11129  nn0lt2  12604  fzofzim  13677  ndvdssub  16386  cyccom  19142  alexsubALT  23945  clwlkclwwlklem2a  29934  erclwwlktr  29958  erclwwlkntr  30007  fmlasuc  35380  dfon2lem8  35785  prtlem15  38875  prtlem18  38877  2reuimp0  47119