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Theorem iuneq12df 4676
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 2835 . . . 4 (𝜑 → (𝑦𝐶𝑦𝐷))
71, 2, 3, 4, 6rexeqbid 3299 . . 3 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
87alrimiv 2006 . 2 (𝜑 → ∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
9 abbi 2885 . . 3 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
10 df-iun 4654 . . . 4 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
11 df-iun 4654 . . . 4 𝑥𝐵 𝐷 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷}
1210, 11eqeq12i 2784 . . 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 1628   = wceq 1630  wnf 1855  wcel 2144  {cab 2756  wnfc 2899  wrex 3061   ciun 4652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-iun 4654
This theorem is referenced by:  iunxdif3  4738  iundisjf  29734  aciunf1  29797  measvuni  30611  iuneq2f  34288
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