Theorem consym1 37188

Description: A symmetry with ∧.

See negsym1 37185 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

Assertion

Ref	Expression
consym1	⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑))

Proof of Theorem consym1

Step	Hyp	Ref	Expression
1		falim 1587	. . 3 ⊢ (⊥ → ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑)))
2	1	ad2antll 742	. 2 ⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑)))
3	2	pm2.43i 53	1 ⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ⊥wfal 1582

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  df-an 402  df-tru 1573  df-fal 1583

This theorem is used by: (None)