Theorem absnsb 48097

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 2742	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2		velsn 4599	. . . . 5 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
3	1, 2	bibi12i 342	. . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑 ↔ 𝑥 = 𝑦))
4		biimpr 223	. . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑))
5	3, 4	sylbi 220	. . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) → (𝑥 = 𝑦 → 𝜑))
6	5	alimi 1843	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
7		nfab1 2924	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
8		nfcv 2922	. . 3 ⊢ Ⅎ𝑥{𝑦}
9	7, 8	cleqf 2950	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}))
10		sb6 2121	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
11	6, 9, 10	3imtr4i 295	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   = wceq 1569  [wsb 2098   ∈ wcel 2144  {cab 2738  {csn 4583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sn 4584

This theorem is used by: (None)