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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 {𝑥𝐴𝜑} = {𝑥𝐵𝜑}

Proof of Theorem rabeqiOLD
StepHypRef Expression
1 rabeqi.1 . 2 𝐴 = 𝐵
21nfth 1803 . . . 4 𝑥 𝐴 = 𝐵
3 eleq2 2840 . . . . 5 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
43anbi1d 632 . . . 4 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
52, 4abbid 2824 . . 3 (𝐴 = 𝐵 → {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐵𝜑)})
6 df-rab 3079 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
7 df-rab 3079 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
85, 6, 73eqtr4g 2818 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
91, 8ax-mp 5 1 {𝑥𝐴𝜑} = {𝑥𝐵𝜑}
 Colors of variables: wff setvar class 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)
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