Description: Obsolete version of rabrabi as of 12-Oct-2024. (Contributed by AV, 2-Aug-2022) Avoid ax-10 and ax-11 . (Revised by Gino Giotto, 20-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | rabrabiOLD.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜒 ↔ 𝜑 ) ) | |
Assertion | rabrabiOLD | ⊢ { 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ ( 𝜒 ∧ 𝜓 ) } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabrabiOLD.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜒 ↔ 𝜑 ) ) | |
2 | 1 | cbvrabv | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜒 } = { 𝑦 ∈ 𝐴 ∣ 𝜑 } |
3 | 2 | rabeqi | ⊢ { 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ∣ 𝜓 } = { 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } ∣ 𝜓 } |
4 | rabrab | ⊢ { 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ ( 𝜒 ∧ 𝜓 ) } | |
5 | 3 4 | eqtr3i | ⊢ { 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ ( 𝜒 ∧ 𝜓 ) } |