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Theorem bj-rspgt 13371
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2813 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
bj-rspg.nfa 𝑥𝐴
bj-rspg.nfb 𝑥𝐵
bj-rspg.nf2 𝑥𝜓
Assertion
Ref Expression
bj-rspgt (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓)))

Proof of Theorem bj-rspgt
StepHypRef Expression
1 eleq1 2220 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
21imbi1d 230 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)) ↔ (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜑))))
32biimpd 143 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜑))))
4 imim2 55 . . . . . . . 8 ((𝜑𝜓) → ((∀𝑥𝐵 𝜑𝜑) → (∀𝑥𝐵 𝜑𝜓)))
54imim2d 54 . . . . . . 7 ((𝜑𝜓) → ((𝐴𝐵 → (∀𝑥𝐵 𝜑𝜑)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))))
63, 5syl9 72 . . . . . 6 (𝑥 = 𝐴 → ((𝜑𝜓) → ((𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))))
76a2i 11 . . . . 5 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → ((𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))))
87alimi 1435 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))))
9 bj-rspg.nfa . . . . 5 𝑥𝐴
10 bj-rspg.nfb . . . . . . 7 𝑥𝐵
119, 10nfel 2308 . . . . . 6 𝑥 𝐴𝐵
12 nfra1 2488 . . . . . . 7 𝑥𝑥𝐵 𝜑
13 bj-rspg.nf2 . . . . . . 7 𝑥𝜓
1412, 13nfim 1552 . . . . . 6 𝑥(∀𝑥𝐵 𝜑𝜓)
1511, 14nfim 1552 . . . . 5 𝑥(𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
16 rsp 2504 . . . . . . 7 (∀𝑥𝐵 𝜑 → (𝑥𝐵𝜑))
1716a1i 9 . . . . . 6 (𝑥 = 𝐴 → (∀𝑥𝐵 𝜑 → (𝑥𝐵𝜑)))
1817com23 78 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)))
199, 15, 18bj-vtoclgft 13360 . . . 4 (∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))) → (𝐴𝐵 → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))))
208, 19syl 14 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))))
2120pm2.43d 50 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))
2221com23 78 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1333   = wceq 1335  wnf 1440  wcel 2128  wnfc 2286  wral 2435
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714
This theorem is referenced by:  bj-rspg  13372
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