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Mirrors > Home > NFE Home > Th. List > cbvexdva | GIF version |
Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
cbvaldva.1 | ⊢ ((φ ∧ x = y) → (ψ ↔ χ)) |
Ref | Expression |
---|---|
cbvexdva | ⊢ (φ → (∃xψ ↔ ∃yχ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . 2 ⊢ Ⅎyφ | |
2 | nfvd 1620 | . 2 ⊢ (φ → Ⅎyψ) | |
3 | cbvaldva.1 | . . 3 ⊢ ((φ ∧ x = y) → (ψ ↔ χ)) | |
4 | 3 | ex 423 | . 2 ⊢ (φ → (x = y → (ψ ↔ χ))) |
5 | 1, 2, 4 | cbvexd 2009 | 1 ⊢ (φ → (∃xψ ↔ ∃yχ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∃wex 1541 |
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-12 1925 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 |
This theorem is referenced by: cbvrexdva2 2840 |
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