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Mirrors > Home > MPE Home > Th. List > rabrabiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rabrabi 3447 as of 12-Oct-2024. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2130 and ax-11 2147. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rabrabiOLD.1 | ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) |
Ref | Expression |
---|---|
rabrabiOLD | ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabrabiOLD.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) | |
2 | 1 | cbvrabv 3439 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑦 ∈ 𝐴 ∣ 𝜑} |
3 | 2 | rabeqi 3442 | . 2 ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ∣ 𝜓} = {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} |
4 | rabrab 3452 | . 2 ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} | |
5 | 3, 4 | eqtr3i 2758 | 1 ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 {crab 3429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 |
This theorem is referenced by: (None) |
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