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Theorem bj-sngleq 36968
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 3322 . . 3 (𝐴 = 𝐵 → (∃𝑦𝐴 𝑥 = {𝑦} ↔ ∃𝑦𝐵 𝑥 = {𝑦}))
21abbidv 2808 . 2 (𝐴 = 𝐵 → {𝑥 ∣ ∃𝑦𝐴 𝑥 = {𝑦}} = {𝑥 ∣ ∃𝑦𝐵 𝑥 = {𝑦}})
3 df-bj-sngl 36967 . 2 sngl 𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥 = {𝑦}}
4 df-bj-sngl 36967 . 2 sngl 𝐵 = {𝑥 ∣ ∃𝑦𝐵 𝑥 = {𝑦}}
52, 3, 43eqtr4g 2802 1 (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  {cab 2714  wrex 3070  {csn 4626  sngl bj-csngl 36966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-rex 3071  df-bj-sngl 36967
This theorem is referenced by:  bj-tageq  36977
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