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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sngleq | Structured version Visualization version GIF version |
Description: Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-sngleq | ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeq 3406 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦})) | |
2 | 1 | abbidv 2885 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}} = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦}}) |
3 | df-bj-sngl 34281 | . 2 ⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}} | |
4 | df-bj-sngl 34281 | . 2 ⊢ sngl 𝐵 = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦}} | |
5 | 2, 3, 4 | 3eqtr4g 2881 | 1 ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 {cab 2799 ∃wrex 3139 {csn 4567 sngl bj-csngl 34280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-rex 3144 df-bj-sngl 34281 |
This theorem is referenced by: bj-tageq 34291 |
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