Theorem rabid2im 3455

Description: One direction of rabid2 3456 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		pm4.71 566	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2	1	albii 1846	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
3		eqab 2907	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
4	2, 3	sylbi 220	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
5		df-ral 3086	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		df-rab 3424	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	6	eqeq2i 2782	. 2 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
8	4, 5, 7	3imtr4i 295	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  {crab 3423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424

This theorem is referenced by:  class2seteq  3676  rabxm  4354