Theorem absnsb 46977

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 2716	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2		velsn 4647	. . . . 5 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
3	1, 2	bibi12i 339	. . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑 ↔ 𝑥 = 𝑦))
4		biimpr 220	. . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑))
5	3, 4	sylbi 217	. . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) → (𝑥 = 𝑦 → 𝜑))
6	5	alimi 1808	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
7		nfab1 2905	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
8		nfcv 2903	. . 3 ⊢ Ⅎ𝑥{𝑦}
9	7, 8	cleqf 2932	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}))
10		sb6 2083	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
11	6, 9, 10	3imtr4i 292	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537  [wsb 2062   ∈ wcel 2106  {cab 2712  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sn 4632

This theorem is referenced by: (None)