Theorem rabid2im 3471

Description: One direction of rabid2 3472 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		pm4.71 557	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2	1	albii 1817	. . 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 217	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
5		df-ral 3064	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		df-rab 3439	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	6	eqeq2i 2747	. 2 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
8	4, 5, 7	3imtr4i 292	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2103  {cab 2711  ∀wral 3063  {crab 3438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rab 3439

This theorem is referenced by:  class2seteq  3720  rabxm  4409