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Theorem ceqsexg 2865
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 2319 . 2 𝑥𝐴
2 nfe1 1496 . . 3 𝑥𝑥(𝑥 = 𝐴𝜑)
3 ceqsexg.1 . . 3 𝑥𝜓
42, 3nfbi 1589 . 2 𝑥(∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
5 ceqex 2864 . . 3 (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
6 ceqsexg.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6bibi12d 235 . 2 (𝑥 = 𝐴 → ((𝜑𝜑) ↔ (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)))
8 biid 171 . 2 (𝜑𝜑)
91, 4, 7, 8vtoclgf 2795 1 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wnf 1460  wex 1492  wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  ceqsexgv  2866
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