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Theorem rspct 2643
 Description: A closed version of rspc 2644. (Contributed by Andrew Salmon, 6-Jun-2011.)
Hypothesis
Ref Expression
rspct.1 xψ
Assertion
Ref Expression
rspct (x(x = A → (φψ)) → (A B → (x B φψ)))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem rspct
StepHypRef Expression
1 df-ral 2305 . . . 4 (x B φx(x Bφ))
2 eleq1 2097 . . . . . . . . . 10 (x = A → (x BA B))
32adantr 261 . . . . . . . . 9 ((x = A (φψ)) → (x BA B))
4 simpr 103 . . . . . . . . 9 ((x = A (φψ)) → (φψ))
53, 4imbi12d 223 . . . . . . . 8 ((x = A (φψ)) → ((x Bφ) ↔ (A Bψ)))
65ex 108 . . . . . . 7 (x = A → ((φψ) → ((x Bφ) ↔ (A Bψ))))
76a2i 11 . . . . . 6 ((x = A → (φψ)) → (x = A → ((x Bφ) ↔ (A Bψ))))
87alimi 1341 . . . . 5 (x(x = A → (φψ)) → x(x = A → ((x Bφ) ↔ (A Bψ))))
9 nfv 1418 . . . . . . 7 x A B
10 rspct.1 . . . . . . 7 xψ
119, 10nfim 1461 . . . . . 6 x(A Bψ)
12 nfcv 2175 . . . . . 6 xA
1311, 12spcgft 2624 . . . . 5 (x(x = A → ((x Bφ) ↔ (A Bψ))) → (A B → (x(x Bφ) → (A Bψ))))
148, 13syl 14 . . . 4 (x(x = A → (φψ)) → (A B → (x(x Bφ) → (A Bψ))))
151, 14syl7bi 154 . . 3 (x(x = A → (φψ)) → (A B → (x B φ → (A Bψ))))
1615com34 77 . 2 (x(x = A → (φψ)) → (A B → (A B → (x B φψ))))
1716pm2.43d 44 1 (x(x = A → (φψ)) → (A B → (x B φψ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  ∀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-bndl 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: (None)
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