Theorem rspcdva 3288

Description: Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Hypotheses

Ref	Expression
rspcdva.1	⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒))
rspcdva.2	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
rspcdva.3	⊢ (𝜑 → 𝐶 ∈ 𝐴)

Assertion

Ref	Expression
rspcdva	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜒,𝑥   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcdva

Step	Hyp	Ref	Expression
1		rspcdva.2	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
2		rspcdva.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
3		rspcdva.1	. . . 4 ⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒))
4	3	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → (𝜓 ↔ 𝜒))
5	2, 4	rspcdv 3285	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → 𝜒))
6	1, 5	mpd 15	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175

This theorem is referenced by:  prmreclem5  15462  gsumzaddlem  18144  ablfac1eu  18295  evlslem1  19336  tayl0  23920  lgamcvglem  24566  inelpisys  29544  unelldsys  29548  ldgenpisyslem1  29553  unblimceq0lem  31667  unblimceq0  31668  unbdqndv2  31672  wlkOnl1iedg  40873  1wlkp1lem7  40888  1wlkp1lem8  40889  crctcsh1wlkn0lem6  41018  eupth2eucrct  41385