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Theorem iuneq2df 39032
 Description: Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
iuneq2df.1 𝑥𝜑
iuneq2df.2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
iuneq2df (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Proof of Theorem iuneq2df
StepHypRef Expression
1 iuneq2df.1 . . 3 𝑥𝜑
2 iuneq2df.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32ex 450 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 2954 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 iuneq2 4528 . 2 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
64, 5syl 17 1 (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481  Ⅎwnf 1706   ∈ wcel 1988  ∀wral 2909  ∪ ciun 4511 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-in 3574  df-ss 3581  df-iun 4513 This theorem is referenced by:  subsaliuncl  40339  omeiunlempt  40497  hoicvrrex  40533  ovnlecvr2  40587  smflimmpt  40779  smflimsupmpt  40798  smfliminfmpt  40801
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