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Theorem iuneq12daf 29692
Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.)
Hypotheses
Ref Expression
iuneq12daf.1 𝑥𝜑
iuneq12daf.2 𝑥𝐴
iuneq12daf.3 𝑥𝐵
iuneq12daf.4 (𝜑𝐴 = 𝐵)
iuneq12daf.5 ((𝜑𝑥𝐴) → 𝐶 = 𝐷)
Assertion
Ref Expression
iuneq12daf (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Proof of Theorem iuneq12daf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iuneq12daf.1 . . . . 5 𝑥𝜑
2 iuneq12daf.5 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶 = 𝐷)
32eleq2d 2867 . . . . 5 ((𝜑𝑥𝐴) → (𝑦𝐶𝑦𝐷))
41, 3rexbida 3231 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐴 𝑦𝐷))
5 iuneq12daf.4 . . . . 5 (𝜑𝐴 = 𝐵)
6 iuneq12daf.2 . . . . . 6 𝑥𝐴
7 iuneq12daf.3 . . . . . 6 𝑥𝐵
86, 7rexeqf 3320 . . . . 5 (𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝐷 ↔ ∃𝑥𝐵 𝑦𝐷))
95, 8syl 17 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦𝐷 ↔ ∃𝑥𝐵 𝑦𝐷))
104, 9bitrd 270 . . 3 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
1110alrimiv 2017 . 2 (𝜑 → ∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
12 abbi 2917 . . 3 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
13 df-iun 4707 . . . 4 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
14 df-iun 4707 . . . 4 𝑥𝐵 𝐷 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷}
1513, 14eqeq12i 2816 . . 3 ( 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
1612, 15bitr4i 269 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) ↔ 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
1711, 16sylib 209 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wnf 1863  wcel 2155  {cab 2788  wnfc 2931  wrex 3093   ciun 4705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-rex 3098  df-iun 4707
This theorem is referenced by:  measvunilem0  30595
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