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Theorem disjeq2dv 3741
Description: Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypothesis
Ref Expression
disjeq2dv.1 ((φ x A) → B = 𝐶)
Assertion
Ref Expression
disjeq2dv (φ → (Disj x A BDisj x A 𝐶))
Distinct variable group:   φ,x
Allowed substitution hints:   A(x)   B(x)   𝐶(x)

Proof of Theorem disjeq2dv
StepHypRef Expression
1 disjeq2dv.1 . . 3 ((φ x A) → B = 𝐶)
21ralrimiva 2386 . 2 (φx A B = 𝐶)
3 disjeq2 3740 . 2 (x A B = 𝐶 → (Disj x A BDisj x A 𝐶))
42, 3syl 14 1 (φ → (Disj x A BDisj x A 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  Disj wdisj 3736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rmo 2308  df-in 2918  df-ss 2925  df-disj 3737
This theorem is referenced by:  disjeq12d  3745
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