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  1453  rspct  3566  zfpair  5390  gruen  10936  axpre-sup  11293  nn0lt2  12802  fzofzim  13884  ndvdssub  16623  cyccom  19468  alexsubALT  24425  clwlkclwwlklem2a  30646  erclwwlktr  30670  erclwwlkntr  30719  fmlasuc  36359  dfon2lem8  36767  prtlem15  40198  prtlem18  40200  2reuimp0  48458