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Theorem eqeqan2d 34322
 Description: Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.)
Hypothesis
Ref Expression
eqeqan2d.1 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
eqeqan2d ((𝐴 = 𝐵𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem eqeqan2d
StepHypRef Expression
1 eqeqan2d.1 . 2 (𝜑𝐶 = 𝐷)
2 eqeq12 2818 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
31, 2sylan2 582 1 ((𝐴 = 𝐵𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1637 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-ext 2784 This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-cleq 2798 This theorem is referenced by: (None)
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