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Theorem disjeq1d 3974
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypothesis
Ref Expression
disjeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
disjeq1d (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem disjeq1d
StepHypRef Expression
1 disjeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 disjeq1 3973 . 2 (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
31, 2syl 14 1 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1348  Disj wdisj 3966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-rmo 2456  df-in 3127  df-ss 3134  df-disj 3967
This theorem is referenced by:  disjeq12d  3975
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