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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvex4vv | Structured version Visualization version GIF version |
Description: Version of cbvex4v 2289 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbvex4vv.1 | ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) |
bj-cbvex4vv.2 | ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
bj-cbvex4vv | ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-cbvex4vv.1 | . . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) | |
2 | 1 | 2exbidv 1852 | . . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤𝜓)) |
3 | 2 | bj-cbvex2vv 32740 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑧∃𝑤𝜓) |
4 | bj-cbvex4vv.2 | . . . 4 ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒)) | |
5 | 4 | bj-cbvex2vv 32740 | . . 3 ⊢ (∃𝑧∃𝑤𝜓 ↔ ∃𝑓∃𝑔𝜒) |
6 | 5 | 2exbii 1775 | . 2 ⊢ (∃𝑣∃𝑢∃𝑧∃𝑤𝜓 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) |
7 | 3, 6 | bitri 264 | 1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: (None) |
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