Theorem rabrabiOLD 3418

Description: Obsolete version of rabrabi 3417 as of 12-Oct-2024. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2139 and ax-11 2156. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
rabrabiOLD.1	⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))

Assertion

Ref	Expression
rabrabiOLD	⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}

Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑥   𝜒,𝑦

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥)

Proof of Theorem rabrabiOLD

Step	Hyp	Ref	Expression
1		rabrabiOLD.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))
2	1	cbvrabv 3416	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑦 ∈ 𝐴 ∣ 𝜑}
3	2	rabeqi 3406	. 2 ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ∣ 𝜓} = {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓}
4		rabrab 3305	. 2 ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}
5	3, 4	eqtr3i 2768	1 ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  {crab 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072

This theorem is referenced by: (None)