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Theorem bj-rspgt 9899
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2653 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 2100 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
21imbi1d 220 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)) ↔ (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜑))))
32biimpd 132 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜑))))
4 imim2 49 . . . . . . . 8 ((𝜑𝜓) → ((∀𝑥𝐵 𝜑𝜑) → (∀𝑥𝐵 𝜑𝜓)))
54imim2d 48 . . . . . . 7 ((𝜑𝜓) → ((𝐴𝐵 → (∀𝑥𝐵 𝜑𝜑)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))))
63, 5syl9 66 . . . . . 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 1344 . . . 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 2186 . . . . . 6 𝑥 𝐴𝐵
12 nfra1 2355 . . . . . . 7 𝑥𝑥𝐵 𝜑
13 bj-rspg.nf2 . . . . . . 7 𝑥𝜓
1412, 13nfim 1464 . . . . . 6 𝑥(∀𝑥𝐵 𝜑𝜓)
1511, 14nfim 1464 . . . . 5 𝑥(𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
16 rsp 2369 . . . . . . 7 (∀𝑥𝐵 𝜑 → (𝑥𝐵𝜑))
1716a1i 9 . . . . . 6 (𝑥 = 𝐴 → (∀𝑥𝐵 𝜑 → (𝑥𝐵𝜑)))
1817com23 72 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)))
199, 15, 18bj-vtoclgft 9888 . . . 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 44 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))
2221com23 72 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1241   = wceq 1243  wnf 1349  wcel 1393  wnfc 2165  wral 2306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559
This theorem is referenced by:  bj-rspg  9900
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