Theorem rabbidaOLD 3461

Description: Obsolete version of rabbida 3447 as of 14-Mar-2025. (Contributed by BJ, 27-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rabbidaOLD.n	⊢ Ⅎ𝑥𝜑
rabbidaOLD.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rabbidaOLD	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Proof of Theorem rabbidaOLD

Step	Hyp	Ref	Expression
1		rabbidaOLD.n	. . 3 ⊢ Ⅎ𝑥𝜑
2		rabbidaOLD.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	ex 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
4	1, 3	ralrimi 3244	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒))
5		rabbi 3451	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
6	4, 5	sylib 218	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  ∀wral 3052  {crab 3420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-ral 3053  df-rab 3421

This theorem is referenced by: (None)