Theorem bi2.04 247

Description: Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
bi2.04	⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜓 → (𝜑 → 𝜒)))

Proof of Theorem bi2.04

Step	Hyp	Ref	Expression
1		pm2.04 82	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
2		pm2.04 82	. 2 ⊢ ((𝜓 → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → 𝜒)))
3	1, 2	impbii 125	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜓 → (𝜑 → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  imim21b  251  pm4.87  525  imimorbdc  834  sbcom2v  1910  mor  1991  r19.21t  2449  reu8  2812  ra5  2928  unissb  3689  reusv3  4295  zfregfr  4402  tfi  4410  fun11  5094  prime  8906  raluz2  9128  isprm3  11439  isprm4  11440  bj-inf2vnlem2  12139