Theorem bnj1322 35318

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 3992	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
2		dfss2 3920	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
3	1, 2	sylib 221	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∩ cin 3901   ⊆ wss 3902

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-in 3909  df-ss 3919

This theorem is used by:  bnj1321  35523