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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elequ12 | Structured version Visualization version GIF version |
Description: An identity law for the non-logical predicate, which combines elequ1 1984 and elequ2 1991. For the analogous theorems for class terms, see eleq1 2676, eleq2 2677 and eleq12 2678. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.) |
Ref | Expression |
---|---|
bj-elequ12 | ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ1 1984 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) | |
2 | elequ2 1991 | . 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡)) | |
3 | 1, 2 | sylan9bb 732 | 1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 |
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-8 1979 ax-9 1986 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: bj-ru0 32124 |
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