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Theorem iotauni2 6522
Description: Version of iotauni 6528 using df-iota 6505 instead of dfiota2 6506. (Contributed by SN, 6-Nov-2024.)
Assertion
Ref Expression
iotauni2 (∃𝑦{𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = {𝑥𝜑})
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem iotauni2
StepHypRef Expression
1 iotaval2 6521 . . 3 ({𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2 unieq 4923 . . . 4 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = {𝑦})
3 unisnv 4934 . . . 4 {𝑦} = 𝑦
42, 3eqtr2di 2785 . . 3 ({𝑥𝜑} = {𝑦} → 𝑦 = {𝑥𝜑})
51, 4eqtrd 2768 . 2 ({𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = {𝑥𝜑})
65exlimiv 1925 1 (∃𝑦{𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wex 1773  {cab 2705  {csn 4632   cuni 4912  cio 6503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-un 3954  df-in 3956  df-ss 3966  df-sn 4633  df-pr 4635  df-uni 4913  df-iota 6505
This theorem is referenced by:  iotassuni  6525
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