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Theorem bj-rspgt 9194
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 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)) → (x B φ → (A Bψ)))

Proof of Theorem bj-rspgt
StepHypRef Expression
1 eleq1 2097 . . . . . . . . 9 (x = A → (x BA B))
21imbi1d 220 . . . . . . . 8 (x = A → ((x B → (x B φφ)) ↔ (A B → (x B φφ))))
32biimpd 132 . . . . . . 7 (x = A → ((x B → (x B φφ)) → (A B → (x B φφ))))
4 imim2 49 . . . . . . . 8 ((φψ) → ((x B φφ) → (x B φψ)))
54imim2d 48 . . . . . . 7 ((φψ) → ((A B → (x B φφ)) → (A B → (x B φψ))))
63, 5syl9 66 . . . . . 6 (x = A → ((φψ) → ((x B → (x B φφ)) → (A B → (x B φψ)))))
76a2i 11 . . . . 5 ((x = A → (φψ)) → (x = A → ((x B → (x B φφ)) → (A B → (x B φψ)))))
87alimi 1341 . . . 4 (x(x = A → (φψ)) → x(x = A → ((x B → (x B φφ)) → (A B → (x B φψ)))))
9 bj-rspg.nfa . . . . 5 xA
10 bj-rspg.nfb . . . . . . 7 xB
119, 10nfel 2183 . . . . . 6 x A B
12 nfra1 2349 . . . . . . 7 xx B φ
13 bj-rspg.nf2 . . . . . . 7 xψ
1412, 13nfim 1461 . . . . . 6 x(x B φψ)
1511, 14nfim 1461 . . . . 5 x(A B → (x B φψ))
16 rsp 2363 . . . . . . 7 (x B φ → (x Bφ))
1716a1i 9 . . . . . 6 (x = A → (x B φ → (x Bφ)))
1817com23 72 . . . . 5 (x = A → (x B → (x B φφ)))
199, 15, 18bj-vtoclgft 9183 . . . 4 (x(x = A → ((x B → (x B φφ)) → (A B → (x B φψ)))) → (A B → (A B → (x B φψ))))
208, 19syl 14 . . 3 (x(x = A → (φψ)) → (A B → (A B → (x B φψ))))
2120pm2.43d 44 . 2 (x(x = A → (φψ)) → (A B → (x B φψ)))
2221com23 72 1 (x(x = A → (φψ)) → (x B φ → (A Bψ)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   = wceq 1242  wnf 1346   wcel 1390  wnfc 2162  wral 2300
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553
This theorem is referenced by:  bj-rspg  9195
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