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Theorem iunxprg 4741
Description: A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
Hypotheses
Ref Expression
iunxprg.1 (𝑥 = 𝐴𝐶 = 𝐷)
iunxprg.2 (𝑥 = 𝐵𝐶 = 𝐸)
Assertion
Ref Expression
iunxprg ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸
Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem iunxprg
StepHypRef Expression
1 df-pr 4319 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2 iuneq1 4668 . . . 4 ({𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶)
31, 2ax-mp 5 . . 3 𝑥 ∈ {𝐴, 𝐵}𝐶 = 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶
4 iunxun 4739 . . 3 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶 = ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶)
53, 4eqtri 2793 . 2 𝑥 ∈ {𝐴, 𝐵}𝐶 = ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶)
6 iunxprg.1 . . . . 5 (𝑥 = 𝐴𝐶 = 𝐷)
76iunxsng 4736 . . . 4 (𝐴𝑉 𝑥 ∈ {𝐴}𝐶 = 𝐷)
87adantr 466 . . 3 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴}𝐶 = 𝐷)
9 iunxprg.2 . . . . 5 (𝑥 = 𝐵𝐶 = 𝐸)
109iunxsng 4736 . . . 4 (𝐵𝑊 𝑥 ∈ {𝐵}𝐶 = 𝐸)
1110adantl 467 . . 3 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐵}𝐶 = 𝐸)
128, 11uneq12d 3919 . 2 ((𝐴𝑉𝐵𝑊) → ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶) = (𝐷𝐸))
135, 12syl5eq 2817 1 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cun 3721  {csn 4316  {cpr 4318   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-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  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-ral 3066  df-rex 3067  df-v 3353  df-sbc 3588  df-un 3728  df-in 3730  df-ss 3737  df-sn 4317  df-pr 4319  df-iun 4656
This theorem is referenced by:  ovnsubadd2lem  41379  rnfdmpr  41823
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