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Theorem bj-rspgt 12795
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2758 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 2178 . . . . . . . . 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 1414 . . . 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 2265 . . . . . 6 𝑥 𝐴𝐵
12 nfra1 2441 . . . . . . 7 𝑥𝑥𝐵 𝜑
13 bj-rspg.nf2 . . . . . . 7 𝑥𝜓
1412, 13nfim 1534 . . . . . 6 𝑥(∀𝑥𝐵 𝜑𝜓)
1511, 14nfim 1534 . . . . 5 𝑥(𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
16 rsp 2455 . . . . . . 7 (∀𝑥𝐵 𝜑 → (𝑥𝐵𝜑))
1716a1i 9 . . . . . 6 (𝑥 = 𝐴 → (∀𝑥𝐵 𝜑 → (𝑥𝐵𝜑)))
1817com23 78 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵 → (∀𝑥𝐵 𝜑𝜑)))
199, 15, 18bj-vtoclgft 12784 . . . 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 1312   = wceq 1314  wnf 1419  wcel 1463  wnfc 2243  wral 2391
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660
This theorem is referenced by:  bj-rspg  12796
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