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Mirrors > Home > MPE Home > Th. List > Mathboxes > axextdfeq | Structured version Visualization version GIF version |
Description: A version of ax-ext 2796 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.) |
Ref | Expression |
---|---|
axextdfeq | ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axextnd 10016 | . . 3 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | |
2 | ax8 2119 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) | |
3 | 2 | imim2i 16 | . . 3 ⊢ (((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) → ((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))) |
4 | 1, 3 | eximii 1836 | . 2 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) |
5 | biimpexp 32950 | . . 3 ⊢ (((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ↔ ((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)))) | |
6 | 5 | exbii 1847 | . 2 ⊢ (∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ↔ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)))) |
7 | 4, 6 | mpbi 232 | 1 ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃wex 1779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-nfc 2966 |
This theorem is referenced by: (None) |
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