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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 3322 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦})) | |
| 2 | 1 | abbidv 2808 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}} = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦}}) |
| 3 | df-bj-sngl 36967 | . 2 ⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}} | |
| 4 | df-bj-sngl 36967 | . 2 ⊢ sngl 𝐵 = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦}} | |
| 5 | 2, 3, 4 | 3eqtr4g 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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