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Theorem elrab3t 2691
 Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2693.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
Assertion
Ref Expression
elrab3t ((x(x = A → (φψ)) A B) → (A {x Bφ} ↔ ψ))
Distinct variable groups:   x,A   x,B   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem elrab3t
StepHypRef Expression
1 simpr 103 . . 3 ((x(x = A → (φψ)) A B) → A B)
2 nfa1 1431 . . . . 5 xx(x = A → (φψ))
3 nfv 1418 . . . . 5 x A B
42, 3nfan 1454 . . . 4 x(x(x = A → (φψ)) A B)
5 simpl 102 . . . . . 6 ((x(x = A → (φψ)) A B) → x(x = A → (φψ)))
6519.21bi 1447 . . . . 5 ((x(x = A → (φψ)) A B) → (x = A → (φψ)))
7 eleq1 2097 . . . . . . . . . 10 (x = A → (x BA B))
87biimparc 283 . . . . . . . . 9 ((A B x = A) → x B)
98biantrurd 289 . . . . . . . 8 ((A B x = A) → (φ ↔ (x B φ)))
109bibi1d 222 . . . . . . 7 ((A B x = A) → ((φψ) ↔ ((x B φ) ↔ ψ)))
1110pm5.74da 417 . . . . . 6 (A B → ((x = A → (φψ)) ↔ (x = A → ((x B φ) ↔ ψ))))
1211adantl 262 . . . . 5 ((x(x = A → (φψ)) A B) → ((x = A → (φψ)) ↔ (x = A → ((x B φ) ↔ ψ))))
136, 12mpbid 135 . . . 4 ((x(x = A → (φψ)) A B) → (x = A → ((x B φ) ↔ ψ)))
144, 13alrimi 1412 . . 3 ((x(x = A → (φψ)) A B) → x(x = A → ((x B φ) ↔ ψ)))
15 elabgt 2678 . . 3 ((A B x(x = A → ((x B φ) ↔ ψ))) → (A {x ∣ (x B φ)} ↔ ψ))
161, 14, 15syl2anc 391 . 2 ((x(x = A → (φψ)) A B) → (A {x ∣ (x B φ)} ↔ ψ))
17 df-rab 2309 . . . 4 {x Bφ} = {x ∣ (x B φ)}
1817eleq2i 2101 . . 3 (A {x Bφ} ↔ A {x ∣ (x B φ)})
1918bibi1i 217 . 2 ((A {x Bφ} ↔ ψ) ↔ (A {x ∣ (x B φ)} ↔ ψ))
2016, 19sylibr 137 1 ((x(x = A → (φψ)) A B) → (A {x Bφ} ↔ ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  {cab 2023  {crab 2304 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-rab 2309  df-v 2553 This theorem is referenced by:  f1oresrab  5270
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