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Theorem elabgt 2678
Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2682.) (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 {xφ} ↔ ψ))
Distinct variable groups:   x,A   ψ,x
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem elabgt
StepHypRef Expression
1 abid 2025 . . . . . . 7 (x {xφ} ↔ φ)
2 eleq1 2097 . . . . . . 7 (x = A → (x {xφ} ↔ A {xφ}))
31, 2syl5bbr 183 . . . . . 6 (x = A → (φA {xφ}))
43bibi1d 222 . . . . 5 (x = A → ((φψ) ↔ (A {xφ} ↔ ψ)))
54biimpd 132 . . . 4 (x = A → ((φψ) → (A {xφ} ↔ ψ)))
65a2i 11 . . 3 ((x = A → (φψ)) → (x = A → (A {xφ} ↔ ψ)))
76alimi 1341 . 2 (x(x = A → (φψ)) → x(x = A → (A {xφ} ↔ ψ)))
8 nfcv 2175 . . . 4 xA
9 nfab1 2177 . . . . . 6 x{xφ}
109nfel2 2187 . . . . 5 x A {xφ}
11 nfv 1418 . . . . 5 xψ
1210, 11nfbi 1478 . . . 4 x(A {xφ} ↔ ψ)
13 pm5.5 231 . . . 4 (x = A → ((x = A → (A {xφ} ↔ ψ)) ↔ (A {xφ} ↔ ψ)))
148, 12, 13spcgf 2629 . . 3 (A B → (x(x = A → (A {xφ} ↔ ψ)) → (A {xφ} ↔ ψ)))
1514imp 115 . 2 ((A B x(x = A → (A {xφ} ↔ ψ))) → (A {xφ} ↔ ψ))
167, 15sylan2 270 1 ((A B x(x = A → (φψ))) → (A {xφ} ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  {cab 2023
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-v 2553
This theorem is referenced by:  elrab3t  2691
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