Theorem syl7bi 244

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 205	. 2 ⊢ (𝜑 → 𝜓)
3		syl7bi.2	. 2 ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏)))
4	2, 3	syl7 72	1 ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏)))

Syntax hints:   → wi 4   ↔ wb 195

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 196

This theorem is referenced by:  rspct  3275  zfpair  4831  gruen  9513  axpre-sup  9869  nn0lt2  11317  fzofzim  12382  ndvdssub  14971  alexsubALT  21665  clwlkisclwwlklem2a  26313  erclwwlktr  26343  erclwwlkntr  26355  dfon2lem8  30939  bj-nfimt  32025  prtlem15  33178  prtlem18  33180  clwlkclwwlklem2a  41207  erclwwlkstr  41243  erclwwlksntr  41255