Theorem eqabdv 2318

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Revised by Wolf Lammen, 6-May-2023.)

Hypothesis

Ref	Expression
eqabdv.1	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))

Assertion

Ref	Expression
eqabdv	⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem eqabdv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqabdv.1	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))
2	1	sbbidv 1896	. . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝜓))
3		clelsb1 2294	. . . 4 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
4	3	bicomi 132	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴)
5		df-clab 2176	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
6	2, 4, 5	3bitr4g 223	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ {𝑥 ∣ 𝜓}))
7	6	eqrdv 2187	1 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  [wsb 1773   ∈ wcel 2160  {cab 2175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185

This theorem is referenced by:  dfrhm2  13504