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Theorem ralsnsg 4513
 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 3732 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
2 df-ral 3110 . . 3 (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝜑))
3 velsn 4488 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
43imbi1i 351 . . . 4 ((𝑥 ∈ {𝐴} → 𝜑) ↔ (𝑥 = 𝐴𝜑))
54albii 1801 . . 3 (∀𝑥(𝑥 ∈ {𝐴} → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑))
62, 5bitri 276 . 2 (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑))
71, 6syl6rbbr 291 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1520   = wceq 1522   ∈ wcel 2081  ∀wral 3105  [wsbc 3706  {csn 4472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-v 3439  df-sbc 3707  df-sn 4473 This theorem is referenced by:  ralsngOLD  4515  ralsngf  4518  ixpsnval  8313  ac6sfi  8608  rexfiuz  14541  prmind2  15858  finixpnum  34408
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