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Theorem rspct 2948
 Description: A closed version of rspc 2949. (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 2619 . . . 4 (x B φx(x Bφ))
2 eleq1 2413 . . . . . . . . . 10 (x = A → (x BA B))
32adantr 451 . . . . . . . . 9 ((x = A (φψ)) → (x BA B))
4 simpr 447 . . . . . . . . 9 ((x = A (φψ)) → (φψ))
53, 4imbi12d 311 . . . . . . . 8 ((x = A (φψ)) → ((x Bφ) ↔ (A Bψ)))
65ex 423 . . . . . . 7 (x = A → ((φψ) → ((x Bφ) ↔ (A Bψ))))
76a2i 12 . . . . . 6 ((x = A → (φψ)) → (x = A → ((x Bφ) ↔ (A Bψ))))
87alimi 1559 . . . . 5 (x(x = A → (φψ)) → x(x = A → ((x Bφ) ↔ (A Bψ))))
9 nfv 1619 . . . . . . 7 x A B
10 rspct.1 . . . . . . 7 xψ
119, 10nfim 1813 . . . . . 6 x(A Bψ)
12 nfcv 2489 . . . . . 6 xA
1311, 12spcgft 2931 . . . . 5 (x(x = A → ((x Bφ) ↔ (A Bψ))) → (A B → (x(x Bφ) → (A Bψ))))
148, 13syl 15 . . . 4 (x(x = A → (φψ)) → (A B → (x(x Bφ) → (A Bψ))))
151, 14syl7bi 221 . . 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 setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2614 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861 This theorem is referenced by: (None)
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