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Theorem iuneq12df 4510
 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 2684 . . . 4 (𝜑 → (𝑦𝐶𝑦𝐷))
71, 2, 3, 4, 6rexeqbid 3140 . . 3 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
87alrimiv 1852 . 2 (𝜑 → ∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
9 abbi 2734 . . 3 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
10 df-iun 4487 . . . 4 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
11 df-iun 4487 . . . 4 𝑥𝐵 𝐷 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷}
1210, 11eqeq12i 2635 . . 3 ( 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
139, 12bitr4i 267 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) ↔ 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
148, 13sylib 208 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478   = wceq 1480  Ⅎwnf 1705   ∈ wcel 1987  {cab 2607  Ⅎwnfc 2748  ∃wrex 2908  ∪ ciun 4485 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-iun 4487 This theorem is referenced by:  iunxdif3  4572  iundisjf  29247  aciunf1  29305  measvuni  30058  iuneq2f  33595
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