Theorem bnj1322 35180

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 4003	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
2		dfss2 3931	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
3	1, 2	sylib 221	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1568   ∩ cin 3912   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-in 3920  df-ss 3930

This theorem is referenced by:  bnj1321  35385