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Theorem bj-elequ12 32310
Description: An identity law for the non-logical predicate, which combines elequ1 1994 and elequ2 2001. For the analogous theorems for class terms, see eleq1 2686, eleq2 2687 and eleq12 2688. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.)
Assertion
Ref Expression
bj-elequ12 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Proof of Theorem bj-elequ12
StepHypRef Expression
1 elequ1 1994 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
2 elequ2 2001 . 2 (𝑧 = 𝑡 → (𝑦𝑧𝑦𝑡))
31, 2sylan9bb 735 1 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  bj-ru0  32579
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