Theorem ralsn 3732

Description: Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)

Hypotheses

Ref	Expression
ralsn.1	⊢ 𝐴 ∈ V
ralsn.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralsn	⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralsn

Step	Hyp	Ref	Expression
1		ralsn.1	. 2 ⊢ 𝐴 ∈ V
2		ralsn.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	ralsng 3729	. 2 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520  Vcvv 2813  {csn 3689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-sbc 3043  df-sn 3695

This theorem is referenced by:  tfr0dm  6553  elixpsn  6970  finomni  7431  hashfibc  11207  eqs1  11316  wlkl1loop  16353  clwwlkn2  16416  nninfsellemdc  16788