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Mirrors > Home > NFE Home > Th. List > ceqsexgv | GIF version |
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) |
Ref | Expression |
---|---|
ceqsexgv.1 | ⊢ (x = A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
ceqsexgv | ⊢ (A ∈ V → (∃x(x = A ∧ φ) ↔ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . 2 ⊢ Ⅎxψ | |
2 | ceqsexgv.1 | . 2 ⊢ (x = A → (φ ↔ ψ)) | |
3 | 1, 2 | ceqsexg 2970 | 1 ⊢ (A ∈ V → (∃x(x = A ∧ φ) ↔ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 |
This theorem is referenced by: ceqsrexv 2972 clel3g 2976 eluni1g 4172 opkelopkabg 4245 otkelins2kg 4253 otkelins3kg 4254 opkelcokg 4261 |
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