Theorem consym1 34751

Description: A symmetry with ∧.

See negsym1 34748 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 1558	. . 3 ⊢ (⊥ → ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑)))
2	1	ad2antll 727	. 2 ⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑)))
3	2	pm2.43i 52	1 ⊢ ((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 397  ⊥wfal 1553

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 206  df-an 398  df-tru 1544  df-fal 1554

This theorem is referenced by: (None)