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Theorem disjeq12d 3915
 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 3914 . 2 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
3 disjeq12d.1 . . . 4 (𝜑𝐶 = 𝐷)
43adantr 274 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
54disjeq2dv 3911 . 2 (𝜑 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 𝐷))
62, 5bitrd 187 1 (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  Disj wdisj 3906 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-rmo 2424  df-in 3077  df-ss 3084  df-disj 3907 This theorem is referenced by: (None)
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