Theorem rexsngf 4637

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) (Revised by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
rexsngf.1	⊢ Ⅎ𝑥𝜓
rexsngf.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexsngf	⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem rexsngf

Step	Hyp	Ref	Expression
1		rexsns 4636	. 2 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)
2		rexsngf.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		rexsngf.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	2, 3	sbciegf 3781	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
5	1, 4	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569  Ⅎwnf 1812   ∈ wcel 2142  ∃wrex 3088  [wsbc 3743  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3456  df-sbc 3744  df-sn 4589

This theorem is used by:  reusngf  4639  rexprgf  4660  rmosn  4684  iunxsngf  5057