Theorem bnj1322 34815

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 4054	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
2		dfss2 3981	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
3	1, 2	sylib 218	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3962   ⊆ wss 3963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-in 3970  df-ss 3980

This theorem is referenced by:  bnj1321  35020