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Theorem bj-rspgt 10312
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2670 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 2116 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
21imbi1d 224 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)) ↔ (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜑))))
32biimpd 136 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜑))))
4 imim2 53 . . . . . . . 8 ((𝜑𝜓) → ((∀𝑥𝐵 𝜑𝜑) → (∀𝑥𝐵 𝜑𝜓)))
54imim2d 52 . . . . . . 7 ((𝜑𝜓) → ((𝐴𝐵 → (∀𝑥𝐵 𝜑𝜑)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))))
63, 5syl9 70 . . . . . 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 1360 . . . 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 2202 . . . . . 6 𝑥 𝐴𝐵
12 nfra1 2372 . . . . . . 7 𝑥𝑥𝐵 𝜑
13 bj-rspg.nf2 . . . . . . 7 𝑥𝜓
1412, 13nfim 1480 . . . . . 6 𝑥(∀𝑥𝐵 𝜑𝜓)
1511, 14nfim 1480 . . . . 5 𝑥(𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
16 rsp 2386 . . . . . . 7 (∀𝑥𝐵 𝜑 → (𝑥𝐵𝜑))
1716a1i 9 . . . . . 6 (𝑥 = 𝐴 → (∀𝑥𝐵 𝜑 → (𝑥𝐵𝜑)))
1817com23 76 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)))
199, 15, 18bj-vtoclgft 10301 . . . 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 48 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))
2221com23 76 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257   = wceq 1259  wnf 1365  wcel 1409  wnfc 2181  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576
This theorem is referenced by:  bj-rspg  10313
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