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Theorem bj-elequ12 33005
Description: An identity law for the non-logical predicate, which combines elequ1 2152 and elequ2 2159. For the analogous theorems for class terms, see eleq1 2838, eleq2 2839 and eleq12 2840. (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 2152 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
2 elequ2 2159 . 2 (𝑧 = 𝑡 → (𝑦𝑧𝑦𝑡))
31, 2sylan9bb 499 1 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by:  bj-ru0  33264
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