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Theorem iuneq2df 39730
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 449 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
41, 3ralrimi 3096 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
5 iuneq2 4690 . 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 383   = wceq 1632  wnf 1857  wcel 2140  wral 3051   ciun 4673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-v 3343  df-in 3723  df-ss 3730  df-iun 4675
This theorem is referenced by:  subsaliuncl  41098  omeiunlempt  41259  hoicvrrex  41295  ovnlecvr2  41349  smflimmpt  41541  smflimsupmpt  41560  smfliminfmpt  41563
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