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Mirrors > Home > NFE Home > Th. List > ceqsex2v | GIF version |
Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
Ref | Expression |
---|---|
ceqsex2v.1 | ⊢ A ∈ V |
ceqsex2v.2 | ⊢ B ∈ V |
ceqsex2v.3 | ⊢ (x = A → (φ ↔ ψ)) |
ceqsex2v.4 | ⊢ (y = B → (ψ ↔ χ)) |
Ref | Expression |
---|---|
ceqsex2v | ⊢ (∃x∃y(x = A ∧ y = B ∧ φ) ↔ χ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . 2 ⊢ Ⅎxψ | |
2 | nfv 1619 | . 2 ⊢ Ⅎyχ | |
3 | ceqsex2v.1 | . 2 ⊢ A ∈ V | |
4 | ceqsex2v.2 | . 2 ⊢ B ∈ V | |
5 | ceqsex2v.3 | . 2 ⊢ (x = A → (φ ↔ ψ)) | |
6 | ceqsex2v.4 | . 2 ⊢ (y = B → (ψ ↔ χ)) | |
7 | 1, 2, 3, 4, 5, 6 | ceqsex2 2895 | 1 ⊢ (∃x∃y(x = A ∧ y = B ∧ φ) ↔ χ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ w3a 934 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-3an 936 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-v 2861 |
This theorem is referenced by: ceqsex3v 2897 ceqsex4v 2898 opksnelsik 4265 sikexlem 4295 br1stg 4730 elswap 4740 brswap2 4860 brsnsi 5773 oqelins4 5794 dmpprod 5840 lecex 6115 addccan2nclem1 6263 nmembers1lem1 6268 |
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