Theorem bj-rspg 15923

Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2881 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 15922	. 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 1475	1 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))

Syntax hints:   → wi 4   = wceq 1373  Ⅎwnf 1484   ∈ wcel 2178  Ⅎwnfc 2337  ∀wral 2486

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778

This theorem is referenced by:  bj-bdfindisg  16083  bj-findisg  16115