Theorem syl7bi 258

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 219	. 2 ⊢ (𝜑 → 𝜓)
3		syl7bi.2	. 2 ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏)))
4	2, 3	syl7 75	1 ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  3jao  1452  rspct  3563  zfpair  5383  gruen  10890  axpre-sup  11247  nn0lt2  12755  fzofzim  13837  ndvdssub  16572  cyccom  19411  alexsubALT  24363  clwlkclwwlklem2a  30582  erclwwlktr  30606  erclwwlkntr  30655  fmlasuc  36130  dfon2lem8  36532  prtlem15  39912  prtlem18  39914  2reuimp0  48153