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Theorem iuneq12daf 32257
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 2811 . . . . 5 ((𝜑𝑥𝐴) → (𝑦𝐶𝑦𝐷))
41, 3rexbida 3261 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐴 𝑦𝐷))
5 iuneq12daf.4 . . . . 5 (𝜑𝐴 = 𝐵)
6 iuneq12daf.2 . . . . . 6 𝑥𝐴
7 iuneq12daf.3 . . . . . 6 𝑥𝐵
86, 7rexeqf 3342 . . . . 5 (𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝐷 ↔ ∃𝑥𝐵 𝑦𝐷))
95, 8syl 17 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦𝐷 ↔ ∃𝑥𝐵 𝑦𝐷))
104, 9bitrd 279 . . 3 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
1110alrimiv 1922 . 2 (𝜑 → ∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
12 abbi 2792 . . 3 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
13 df-iun 4989 . . 3 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
14 df-iun 4989 . . 3 𝑥𝐵 𝐷 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷}
1512, 13, 143eqtr4g 2789 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
1611, 15syl 17 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  wnf 1777  wcel 2098  {cab 2701  wnfc 2875  wrex 3062   ciun 4987
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 2695
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 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-iun 4989
This theorem is referenced by:  measvunilem0  33700
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