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Theorem elabgt 2982
 Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2986.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
elabgt ((A B x(x = A → (φψ))) → (A {x φ} ↔ ψ))
Distinct variable groups:   x,A   ψ,x
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem elabgt
StepHypRef Expression
1 abid 2341 . . . . . . 7 (x {x φ} ↔ φ)
2 eleq1 2413 . . . . . . 7 (x = A → (x {x φ} ↔ A {x φ}))
31, 2syl5bbr 250 . . . . . 6 (x = A → (φA {x φ}))
43bibi1d 310 . . . . 5 (x = A → ((φψ) ↔ (A {x φ} ↔ ψ)))
54biimpd 198 . . . 4 (x = A → ((φψ) → (A {x φ} ↔ ψ)))
65a2i 12 . . 3 ((x = A → (φψ)) → (x = A → (A {x φ} ↔ ψ)))
76alimi 1559 . 2 (x(x = A → (φψ)) → x(x = A → (A {x φ} ↔ ψ)))
8 nfcv 2489 . . . 4 xA
9 nfab1 2491 . . . . . 6 x{x φ}
109nfel2 2501 . . . . 5 x A {x φ}
11 nfv 1619 . . . . 5 xψ
1210, 11nfbi 1834 . . . 4 x(A {x φ} ↔ ψ)
13 pm5.5 326 . . . 4 (x = A → ((x = A → (A {x φ} ↔ ψ)) ↔ (A {x φ} ↔ ψ)))
148, 12, 13spcgf 2934 . . 3 (A B → (x(x = A → (A {x φ} ↔ ψ)) → (A {x φ} ↔ ψ)))
1514imp 418 . 2 ((A B x(x = A → (A {x φ} ↔ ψ))) → (A {x φ} ↔ ψ))
167, 15sylan2 460 1 ((A B x(x = A → (φψ))) → (A {x φ} ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-v 2861 This theorem is referenced by: (None)
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