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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax8dfeq | Structured version Visualization version GIF version |
Description: A version of ax-8 2112 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.) |
Ref | Expression |
---|---|
ax8dfeq | ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2385 | . 2 ⊢ ∃𝑧 𝑧 = 𝑤 | |
2 | ax8 2116 | . . . 4 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 → 𝑧 ∈ 𝑥)) | |
3 | 2 | equcoms 2027 | . . 3 ⊢ (𝑧 = 𝑤 → (𝑤 ∈ 𝑥 → 𝑧 ∈ 𝑥)) |
4 | ax8 2116 | . . 3 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝑦 → 𝑤 ∈ 𝑦)) | |
5 | 3, 4 | imim12d 81 | . 2 ⊢ (𝑧 = 𝑤 → ((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦))) |
6 | 1, 5 | eximii 1843 | 1 ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-12 2175 ax-13 2374 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 |
This theorem is referenced by: (None) |
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