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Theorem iunxprg 5101
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 4634 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2 iuneq1 5013 . . . 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 5099 . . 3 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶 = ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶)
53, 4eqtri 2763 . 2 𝑥 ∈ {𝐴, 𝐵}𝐶 = ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶)
6 iunxprg.1 . . . . 5 (𝑥 = 𝐴𝐶 = 𝐷)
76iunxsng 5095 . . . 4 (𝐴𝑉 𝑥 ∈ {𝐴}𝐶 = 𝐷)
87adantr 480 . . 3 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴}𝐶 = 𝐷)
9 iunxprg.2 . . . . 5 (𝑥 = 𝐵𝐶 = 𝐸)
109iunxsng 5095 . . . 4 (𝐵𝑊 𝑥 ∈ {𝐵}𝐶 = 𝐸)
1110adantl 481 . . 3 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐵}𝐶 = 𝐸)
128, 11uneq12d 4179 . 2 ((𝐴𝑉𝐵𝑊) → ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶) = (𝐷𝐸))
135, 12eqtrid 2787 1 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cun 3961  {csn 4631  {cpr 4633   ciun 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-iun 4998
This theorem is referenced by:  iunxunpr  32588  ovnsubadd2lem  46601  rnfdmpr  47231
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