Theorem rabid2im 3423

Description: One direction of rabid2 3424 is based on fewer axioms. (Contributed by Wolf Lammen, 26-May-2025.)

Assertion

Ref	Expression
rabid2im	⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑})

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabid2im

Step	Hyp	Ref	Expression
1		pm4.71 562	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2	1	albii 1826	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
3		eqab 2877	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
4	2, 3	sylbi 218	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
5		df-ral 3054	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		df-rab 3392	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	6	eqeq2i 2752	. 2 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
8	4, 5, 7	3imtr4i 293	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  {crab 3391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392

This theorem is referenced by:  class2seteq  3645  rabxm  4319