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Theorem iuneq2df 44221
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 412 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 3246 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 iuneq2 5006 . 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 395   = wceq 1533  wnf 1777  wcel 2098  wral 3053   ciun 4987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-in 3947  df-ss 3957  df-iun 4989
This theorem is referenced by:  subsaliuncl  45559  omeiunlempt  45721  hoicvrrex  45757  ovnlecvr2  45811  smflimmpt  46011  smflimsupmpt  46030  smfliminfmpt  46033
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