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Theorem ralsnsg 3407
 Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
ralsnsg (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem ralsnsg
StepHypRef Expression
1 sbc6g 2788 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
2 df-ral 2311 . . 3 (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝜑))
3 velsn 3392 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
43imbi1i 227 . . . 4 ((𝑥 ∈ {𝐴} → 𝜑) ↔ (𝑥 = 𝐴𝜑))
54albii 1359 . . 3 (∀𝑥(𝑥 ∈ {𝐴} → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑))
62, 5bitri 173 . 2 (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑))
71, 6syl6rbbr 188 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  ∀wral 2306  [wsbc 2764  {csn 3375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765  df-sn 3381 This theorem is referenced by:  ac6sfi  6352
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