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Theorem iuneq12df 4734
Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)
Hypotheses
Ref Expression
iuneq12df.1 𝑥𝜑
iuneq12df.2 𝑥𝐴
iuneq12df.3 𝑥𝐵
iuneq12df.4 (𝜑𝐴 = 𝐵)
iuneq12df.5 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
iuneq12df (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Proof of Theorem iuneq12df
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iuneq12df.1 . . . 4 𝑥𝜑
2 iuneq12df.2 . . . 4 𝑥𝐴
3 iuneq12df.3 . . . 4 𝑥𝐵
4 iuneq12df.4 . . . 4 (𝜑𝐴 = 𝐵)
5 iuneq12df.5 . . . . 5 (𝜑𝐶 = 𝐷)
65eleq2d 2864 . . . 4 (𝜑 → (𝑦𝐶𝑦𝐷))
71, 2, 3, 4, 6rexeqbid 3334 . . 3 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
87alrimiv 2023 . 2 (𝜑 → ∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
9 abbi 2914 . . 3 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
10 df-iun 4712 . . . 4 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
11 df-iun 4712 . . . 4 𝑥𝐵 𝐷 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷}
1210, 11eqeq12i 2813 . . 3 ( 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
139, 12bitr4i 270 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) ↔ 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
148, 13sylib 210 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651   = wceq 1653  wnf 1879  wcel 2157  {cab 2785  wnfc 2928  wrex 3090   ciun 4710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-iun 4712
This theorem is referenced by:  iunxdif3  4797  iundisjf  29919  aciunf1  29982  measvuni  30793  iuneq2f  34449
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