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Mirrors > Home > MPE Home > Th. List > rabeqiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rabeqi 3392 as of 3-Jun-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2143 and ax-11 2159. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rabeqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
rabeqiOLD | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | 1 | nfth 1804 | . . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
3 | eleq2 2839 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
4 | 3 | anbi1d 633 | . . . 4 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
5 | 2, 4 | abbid 2825 | . . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
6 | df-rab 3077 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
7 | df-rab 3077 | . . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
8 | 5, 6, 7 | 3eqtr4g 2819 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
9 | 1, 8 | ax-mp 5 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 400 = wceq 1539 ∈ wcel 2112 {cab 2736 {crab 3072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-12 2176 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-rab 3077 |
This theorem is referenced by: (None) |
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