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Theorem bnj1322 31019
 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 → (𝐴𝐵) = 𝐴)

Proof of Theorem bnj1322
StepHypRef Expression
1 eqimss 3690 . 2 (𝐴 = 𝐵𝐴𝐵)
2 df-ss 3621 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
31, 2sylib 208 1 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∩ cin 3606   ⊆ wss 3607 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-in 3614  df-ss 3621 This theorem is referenced by:  bnj1321  31221
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