Theorem absnsb 47626

Description: If the class abstraction {𝑥 ∣ 𝜑} associated with the wff 𝜑 is a singleton, the wff is true for the singleton element. (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
absnsb	⊢ ({𝑥 ∣ 𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem absnsb

Step	Hyp	Ref	Expression
1		abid 2746	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2		velsn 4600	. . . . 5 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
3	1, 2	bibi12i 341	. . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑 ↔ 𝑥 = 𝑦))
4		biimpr 222	. . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑))
5	3, 4	sylbi 219	. . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) → (𝑥 = 𝑦 → 𝜑))
6	5	alimi 1833	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
7		nfab1 2928	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
8		nfcv 2926	. . 3 ⊢ Ⅎ𝑥{𝑦}
9	7, 8	cleqf 2954	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}))
10		sb6 2120	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
11	6, 9, 10	3imtr4i 294	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560   = wceq 1562  [wsb 2092   ∈ wcel 2144  {cab 2742  {csn 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-v 3458  df-sn 4585

This theorem is referenced by: (None)