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Theorem iuneqconst 4897
 Description: Indexed union of identical classes. (Contributed by AV, 5-Mar-2024.)
Hypothesis
Ref Expression
iuneqconst.p (𝑥 = 𝑋𝐵 = 𝐶)
Assertion
Ref Expression
iuneqconst ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → 𝑥𝐴 𝐵 = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝑋
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iuneqconst
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4890 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 iuneqconst.p . . . . . . . 8 (𝑥 = 𝑋𝐵 = 𝐶)
32eleq2d 2837 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝐵𝑦𝐶))
43rspcev 3543 . . . . . 6 ((𝑋𝐴𝑦𝐶) → ∃𝑥𝐴 𝑦𝐵)
54adantlr 714 . . . . 5 (((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) ∧ 𝑦𝐶) → ∃𝑥𝐴 𝑦𝐵)
65ex 416 . . . 4 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
7 nfv 1915 . . . . . 6 𝑥 𝑋𝐴
8 nfra1 3147 . . . . . 6 𝑥𝑥𝐴 𝐵 = 𝐶
97, 8nfan 1900 . . . . 5 𝑥(𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶)
10 nfv 1915 . . . . 5 𝑥 𝑦𝐶
11 rsp 3134 . . . . . . 7 (∀𝑥𝐴 𝐵 = 𝐶 → (𝑥𝐴𝐵 = 𝐶))
12 eleq2 2840 . . . . . . . 8 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
1312biimpd 232 . . . . . . 7 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
1411, 13syl6 35 . . . . . 6 (∀𝑥𝐴 𝐵 = 𝐶 → (𝑥𝐴 → (𝑦𝐵𝑦𝐶)))
1514adantl 485 . . . . 5 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴 → (𝑦𝐵𝑦𝐶)))
169, 10, 15rexlimd 3241 . . . 4 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (∃𝑥𝐴 𝑦𝐵𝑦𝐶))
176, 16impbid 215 . . 3 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑦𝐶 ↔ ∃𝑥𝐴 𝑦𝐵))
181, 17bitr4id 293 . 2 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
1918eqrdv 2756 1 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → 𝑥𝐴 𝐵 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  ∪ ciun 4886 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-iun 4888 This theorem is referenced by:  uniimafveqt  44294  imasetpreimafvbijlemfv  44315
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