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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 3320 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦})) | |
2 | 1 | abbidv 2806 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}} = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦}}) |
3 | df-bj-sngl 36949 | . 2 ⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}} | |
4 | df-bj-sngl 36949 | . 2 ⊢ sngl 𝐵 = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦}} | |
5 | 2, 3, 4 | 3eqtr4g 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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