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Theorem ceqsexg 2695
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 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsexg
StepHypRef Expression
1 nfcv 2194 . 2 𝑥𝐴
2 nfe1 1401 . . 3 𝑥𝑥(𝑥 = 𝐴𝜑)
3 ceqsexg.1 . . 3 𝑥𝜓
42, 3nfbi 1497 . 2 𝑥(∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
5 ceqex 2694 . . 3 (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
6 ceqsexg.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6bibi12d 228 . 2 (𝑥 = 𝐴 → ((𝜑𝜑) ↔ (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)))
8 biid 164 . 2 (𝜑𝜑)
91, 4, 7, 8vtoclgf 2629 1 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wnf 1365  wex 1397  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  ceqsexgv  2696
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