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Theorem bj-sngleq 32939
Description: Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-sngleq (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)

Proof of Theorem bj-sngleq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 3137 . . 3 (𝐴 = 𝐵 → (∃𝑦𝐴 𝑥 = {𝑦} ↔ ∃𝑦𝐵 𝑥 = {𝑦}))
21abbidv 2740 . 2 (𝐴 = 𝐵 → {𝑥 ∣ ∃𝑦𝐴 𝑥 = {𝑦}} = {𝑥 ∣ ∃𝑦𝐵 𝑥 = {𝑦}})
3 df-bj-sngl 32938 . 2 sngl 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥 = {𝑦}}
4 df-bj-sngl 32938 . 2 sngl 𝐵 = {𝑥 ∣ ∃𝑦𝐵 𝑥 = {𝑦}}
52, 3, 43eqtr4g 2680 1 (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  {cab 2607  wrex 2912  {csn 4175  sngl bj-csngl 32937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-bj-sngl 32938
This theorem is referenced by:  bj-tageq  32948
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