Theorem bj-rspg 13668

Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2827 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
bj-rspg.nfa	⊢ Ⅎ𝑥𝐴
bj-rspg.nfb	⊢ Ⅎ𝑥𝐵
bj-rspg.nf2	⊢ Ⅎ𝑥𝜓
bj-rspg.is	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-rspg	⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))

Proof of Theorem bj-rspg

Step	Hyp	Ref	Expression
1		bj-rspg.nfa	. . 3 ⊢ Ⅎ𝑥𝐴
2		bj-rspg.nfb	. . 3 ⊢ Ⅎ𝑥𝐵
3		bj-rspg.nf2	. . 3 ⊢ Ⅎ𝑥𝜓
4	1, 2, 3	bj-rspgt 13667	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)))
5		bj-rspg.is	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	4, 5	mpg 1439	1 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))

Syntax hints:   → wi 4   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  Ⅎwnfc 2295  ∀wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728

This theorem is referenced by:  bj-bdfindisg  13830  bj-findisg  13862