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Theorem eqeqan1dOLD 2844
 Description: Implication of introducing a new equality. Obsolete as of 14-Feb-2023. Use eqeqan12d 2841 instead. (Contributed by Peter Mazsa, 17-Apr-2019.) (Proof shortened by AV, 10-Feb-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
eqeqan1dOLD.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
eqeqan1dOLD ((𝜑𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem eqeqan1dOLD
StepHypRef Expression
1 eqeqan1dOLD.1 . 2 (𝜑𝐴 = 𝐵)
2 id 22 . 2 (𝐶 = 𝐷𝐶 = 𝐷)
31, 2eqeqan12d 2841 1 ((𝜑𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-cleq 2818 This theorem is referenced by: (None)
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