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Mirrors > Home > MPE Home > Th. List > rabeqif | Structured version Visualization version GIF version |
Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 3190. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
rabeqf.1 | ⊢ Ⅎ𝑥𝐴 |
rabeqf.2 | ⊢ Ⅎ𝑥𝐵 |
rabeqif.3 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
rabeqif | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqif.3 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | rabeqf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | rabeqf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | rabeqf 3190 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 Ⅎwnfc 2751 {crab 2916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 |
This theorem is referenced by: rabeqi 3193 smfliminf 41037 |
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