Theorem absnsb 47502

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 2723	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2		velsn 4573	. . . . 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 1819	. 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 2931	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦}))
10		sb6 2097	. 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 1546   = wceq 1548  [wsb 2074   ∈ wcel 2121  {cab 2719  {csn 4557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-v 3435  df-sn 4558

This theorem is referenced by: (None)