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Theorem disjeq1i 36134
Description: Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.)
Hypothesis
Ref Expression
disjeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
disjeq1i (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶)

Proof of Theorem disjeq1i
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 disjeq1i.1 . . . 4 𝐴 = 𝐵
21rmoeqi 36129 . . 3 (∃*𝑥𝐴 𝑡𝐶 ↔ ∃*𝑥𝐵 𝑡𝐶)
32albii 1814 . 2 (∀𝑡∃*𝑥𝐴 𝑡𝐶 ↔ ∀𝑡∃*𝑥𝐵 𝑡𝐶)
4 df-disj 5118 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑡∃*𝑥𝐴 𝑡𝐶)
5 df-disj 5118 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑡∃*𝑥𝐵 𝑡𝐶)
63, 4, 53bitr4i 303 1 (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1533   = wceq 1535  wcel 2104  ∃*wrmo 3375  Disj wdisj 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-mo 2536  df-cleq 2725  df-clel 2812  df-rmo 3376  df-disj 5118
This theorem is referenced by:  disjeq12i  36135
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