Theorem bnj1322 34453

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1322	⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)

Proof of Theorem bnj1322

Step	Hyp	Ref	Expression
1		eqimss 4038	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
2		df-ss 3964	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
3	1, 2	sylib 217	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1534   ∩ cin 3946   ⊆ wss 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964

This theorem is referenced by:  bnj1321  34658