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Theorem iuneq2df 45052
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 3257 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 iuneq2 5011 . 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 1540  wnf 1783  wcel 2108  wral 3061   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-iun 4993
This theorem is referenced by:  subsaliuncl  46373  omeiunlempt  46535  hoicvrrex  46571  ovnlecvr2  46625  smflimmpt  46825  smflimsupmpt  46844  smfliminfmpt  46847
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