Theorem rabeqiOLD 3395

Description: Obsolete version of rabeqi 3394 as of 3-Jun-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2142 and ax-11 2158. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
rabeqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
rabeqiOLD	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

Proof of Theorem rabeqiOLD

Step	Hyp	Ref	Expression
1		rabeqi.1	. 2 ⊢ 𝐴 = 𝐵
2	1	nfth 1803	. . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐵
3		eleq2 2840	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
4	3	anbi1d 632	. . . 4 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
5	2, 4	abbid 2824	. . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
6		df-rab 3079	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7		df-rab 3079	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
8	5, 6, 7	3eqtr4g 2818	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
9	1, 8	ax-mp 5	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  {crab 3074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079

This theorem is referenced by: (None)