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Theorem iuneq12df 5016
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 2813 . . . 4 (𝜑 → (𝑦𝐶𝑦𝐷))
71, 2, 3, 4, 6rexeqbid 3347 . . 3 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
87alrimiv 1922 . 2 (𝜑 → ∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
9 abbi 2794 . . 3 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
10 df-iun 4992 . . 3 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
11 df-iun 4992 . . 3 𝑥𝐵 𝐷 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷}
129, 10, 113eqtr4g 2791 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
138, 12syl 17 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  wnf 1777  wcel 2098  {cab 2703  wnfc 2877  wrex 3064   ciun 4990
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-iun 4992
This theorem is referenced by:  iunxdif3  5091  iundisjf  32329  aciunf1  32397  measvuni  33742  iuneq2f  37537
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