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Theorem disjeq12d 3986
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disjeq1d.1 (𝜑𝐴 = 𝐵)
disjeq12d.1 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
disjeq12d (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem disjeq12d
StepHypRef Expression
1 disjeq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21disjeq1d 3985 . 2 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
3 disjeq12d.1 . . . 4 (𝜑𝐶 = 𝐷)
43adantr 276 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
54disjeq2dv 3982 . 2 (𝜑 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 𝐷))
62, 5bitrd 188 1 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  Disj wdisj 3977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-rmo 2463  df-in 3135  df-ss 3142  df-disj 3978
This theorem is referenced by: (None)
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