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Theorem bj-sngleq 33438
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 3320 . . 3 (𝐴 = 𝐵 → (∃𝑦𝐴 𝑥 = {𝑦} ↔ ∃𝑦𝐵 𝑥 = {𝑦}))
21abbidv 2916 . 2 (𝐴 = 𝐵 → {𝑥 ∣ ∃𝑦𝐴 𝑥 = {𝑦}} = {𝑥 ∣ ∃𝑦𝐵 𝑥 = {𝑦}})
3 df-bj-sngl 33437 . 2 sngl 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥 = {𝑦}}
4 df-bj-sngl 33437 . 2 sngl 𝐵 = {𝑥 ∣ ∃𝑦𝐵 𝑥 = {𝑦}}
52, 3, 43eqtr4g 2856 1 (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  {cab 2783  wrex 3088  {csn 4366  sngl bj-csngl 33436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rex 3093  df-bj-sngl 33437
This theorem is referenced by:  bj-tageq  33447
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