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Theorem ceqsalg 2576
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
ceqsalg.1 xψ
ceqsalg.2 (x = A → (φψ))
Assertion
Ref Expression
ceqsalg (A 𝑉 → (x(x = Aφ) ↔ ψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)   𝑉(x)

Proof of Theorem ceqsalg
StepHypRef Expression
1 elisset 2562 . . 3 (A 𝑉x x = A)
2 nfa1 1431 . . . 4 xx(x = Aφ)
3 ceqsalg.1 . . . 4 xψ
4 ceqsalg.2 . . . . . . 7 (x = A → (φψ))
54biimpd 132 . . . . . 6 (x = A → (φψ))
65a2i 11 . . . . 5 ((x = Aφ) → (x = Aψ))
76sps 1427 . . . 4 (x(x = Aφ) → (x = Aψ))
82, 3, 7exlimd 1485 . . 3 (x(x = Aφ) → (x x = Aψ))
91, 8syl5com 26 . 2 (A 𝑉 → (x(x = Aφ) → ψ))
104biimprcd 149 . . 3 (ψ → (x = Aφ))
113, 10alrimi 1412 . 2 (ψx(x = Aφ))
129, 11impbid1 130 1 (A 𝑉 → (x(x = Aφ) ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242  wnf 1346  wex 1378   wcel 1390
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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  ceqsal  2577  sbc6g  2782  uniiunlem  3022  sucprcreg  4227  funimass4  5167  ralrnmpt2  5557
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