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Theorem iuneq12df 4919
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 2896 . . . 4 (𝜑 → (𝑦𝐶𝑦𝐷))
71, 2, 3, 4, 6rexeqbid 3403 . . 3 (𝜑 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
87alrimiv 1928 . 2 (𝜑 → ∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷))
9 abbi1 2883 . . 3 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷})
10 df-iun 4895 . . 3 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
11 df-iun 4895 . . 3 𝑥𝐵 𝐷 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐷}
129, 10, 113eqtr4g 2880 . 2 (∀𝑦(∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
138, 12syl 17 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1535   = wceq 1537  wnf 1784  wcel 2114  {cab 2798  wnfc 2957  wrex 3126   ciun 4893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rex 3131  df-iun 4895
This theorem is referenced by:  iunxdif3  4991  iundisjf  30323  aciunf1  30392  measvuni  31478  iuneq2f  35467
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