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Theorem iuneq12daf 29711
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 2836 . . . . 5 ((𝜑𝑥𝐴) → (𝑦𝐶𝑦𝐷))
41, 3rexbida 3195 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐴 𝑦𝐷))
5 iuneq12daf.4 . . . . 5 (𝜑𝐴 = 𝐵)
6 iuneq12daf.2 . . . . . 6 𝑥𝐴
7 iuneq12daf.3 . . . . . 6 𝑥𝐵
86, 7rexeqf 3284 . . . . 5 (𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝐷 ↔ ∃𝑥𝐵 𝑦𝐷))
95, 8syl 17 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦𝐷 ↔ ∃𝑥𝐵 𝑦𝐷))
104, 9bitrd 268 . . 3 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
1110alrimiv 2007 . 2 (𝜑 → ∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
12 abbi 2886 . . 3 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
13 df-iun 4656 . . . 4 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
14 df-iun 4656 . . . 4 𝑥𝐵 𝐷 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷}
1513, 14eqeq12i 2785 . . 3 ( 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
1612, 15bitr4i 267 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) ↔ 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
1711, 16sylib 208 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wnf 1856  wcel 2145  {cab 2757  wnfc 2900  wrex 3062   ciun 4654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-iun 4656
This theorem is referenced by:  measvunilem0  30616
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