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Mirrors > Home > MPE Home > Th. List > cbvexdvaOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of cbvexdva 2271 as of 18-Jul-2021. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvaldva.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvexdvaOLD | ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfvd 1831 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
3 | cbvaldva.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
4 | 3 | ex 449 | . 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
5 | 1, 2, 4 | cbvexd 2266 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: (None) |
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