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Theorem bj-sngleq 36950
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 2806 . 2 (𝐴 = 𝐵 → {𝑥 ∣ ∃𝑦𝐴 𝑥 = {𝑦}} = {𝑥 ∣ ∃𝑦𝐵 𝑥 = {𝑦}})
3 df-bj-sngl 36949 . 2 sngl 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥 = {𝑦}}
4 df-bj-sngl 36949 . 2 sngl 𝐵 = {𝑥 ∣ ∃𝑦𝐵 𝑥 = {𝑦}}
52, 3, 43eqtr4g 2800 1 (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  {cab 2712  wrex 3068  {csn 4631  sngl bj-csngl 36948
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-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-rex 3069  df-bj-sngl 36949
This theorem is referenced by:  bj-tageq  36959
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