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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax8 | Structured version Visualization version GIF version |
Description: Proof of ax-8 1992 from df-clel 2618 (and FOL). This shows that df-clel 2618 is "too powerful". A possible definition is given by bj-df-clel 32888. (Contributed by BJ, 27-Jun-2019.) Also a direct consequence of eleq1w 2684, which has essentially the same proof. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ax8 | ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ2 1953 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑢 = 𝑥 ↔ 𝑢 = 𝑦)) | |
2 | 1 | anbi1d 741 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑢 = 𝑥 ∧ 𝑢 ∈ 𝑧) ↔ (𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧))) |
3 | 2 | exbidv 1850 | . . 3 ⊢ (𝑥 = 𝑦 → (∃𝑢(𝑢 = 𝑥 ∧ 𝑢 ∈ 𝑧) ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧))) |
4 | df-clel 2618 | . . 3 ⊢ (𝑥 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑥 ∧ 𝑢 ∈ 𝑧)) | |
5 | df-clel 2618 | . . 3 ⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧)) | |
6 | 3, 4, 5 | 3bitr4g 303 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
7 | 6 | biimpd 219 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-clel 2618 |
This theorem is referenced by: (None) |
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