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Theorem ceqsexg 2666
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
Hypotheses
Ref Expression
ceqsexg.1 xψ
ceqsexg.2 (x = A → (φψ))
Assertion
Ref Expression
ceqsexg (A 𝑉 → (x(x = A φ) ↔ ψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)   𝑉(x)

Proof of Theorem ceqsexg
StepHypRef Expression
1 nfcv 2175 . 2 xA
2 nfe1 1382 . . 3 xx(x = A φ)
3 ceqsexg.1 . . 3 xψ
42, 3nfbi 1478 . 2 x(x(x = A φ) ↔ ψ)
5 ceqex 2665 . . 3 (x = A → (φx(x = A φ)))
6 ceqsexg.2 . . 3 (x = A → (φψ))
75, 6bibi12d 224 . 2 (x = A → ((φφ) ↔ (x(x = A φ) ↔ ψ)))
8 biid 160 . 2 (φφ)
91, 4, 7, 8vtoclgf 2606 1 (A 𝑉 → (x(x = A φ) ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = 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-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-bndl 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:  ceqsexgv  2667
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