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Theorem iuneq2df 40024
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 403 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 3166 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 iuneq2 4759 . 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 386   = wceq 1656  wnf 1882  wcel 2164  wral 3117   ciun 4742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-in 3805  df-ss 3812  df-iun 4744
This theorem is referenced by:  subsaliuncl  41361  omeiunlempt  41522  hoicvrrex  41558  ovnlecvr2  41612  smflimmpt  41804  smflimsupmpt  41823  smfliminfmpt  41826
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