Theorem bnj1322 35019

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 3975	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
2		dfss2 3903	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
3	1, 2	sylib 220	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1548   ∩ cin 3884   ⊆ wss 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-3an 1095  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-in 3892  df-ss 3902

This theorem is referenced by:  bnj1321  35224