Theorem syl7bi 243

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 204	. 2 ⊢ (𝜑 → 𝜓)
3		syl7bi.2	. 2 ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏)))
4	2, 3	syl7 71	1 ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏)))

Syntax hints:   → wi 4   ↔ wb 194

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 195

This theorem is referenced by:  rspct  3271  zfpair  4823  gruen  9487  axpre-sup  9843  nn0lt2  11270  fzofzim  12334  ndvdssub  14914  alexsubALT  21604  clwlkisclwwlklem2a  26076  erclwwlktr  26106  erclwwlkntr  26118  dfon2lem8  30742  prtlem15  32978  prtlem18  32980  clwlkclwwlklem2a  41206  erclwwlkstr  41242  erclwwlksntr  41254