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Theorem iuneq12daf 30320
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 2875 . . . . 5 ((𝜑𝑥𝐴) → (𝑦𝐶𝑦𝐷))
41, 3rexbida 3277 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐴 𝑦𝐷))
5 iuneq12daf.4 . . . . 5 (𝜑𝐴 = 𝐵)
6 iuneq12daf.2 . . . . . 6 𝑥𝐴
7 iuneq12daf.3 . . . . . 6 𝑥𝐵
86, 7rexeqf 3351 . . . . 5 (𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝐷 ↔ ∃𝑥𝐵 𝑦𝐷))
95, 8syl 17 . . . 4 (𝜑 → (∃𝑥𝐴 𝑦𝐷 ↔ ∃𝑥𝐵 𝑦𝐷))
104, 9bitrd 282 . . 3 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
1110alrimiv 1928 . 2 (𝜑 → ∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
12 abbi1 2861 . . 3 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
13 df-iun 4883 . . 3 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
14 df-iun 4883 . . 3 𝑥𝐵 𝐷 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷}
1512, 13, 143eqtr4g 2858 . 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 209  wa 399  wal 1536   = wceq 1538  wnf 1785  wcel 2111  {cab 2776  wnfc 2936  wrex 3107   ciun 4881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rex 3112  df-iun 4883
This theorem is referenced by:  measvunilem0  31582
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