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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 2114 and elequ2 2122. For the analogous theorems for class terms, see eleq1 2824, eleq2 2825 and eleq12 2826. 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 2114 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) | |
2 | elequ2 2122 | . 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡)) | |
3 | 1, 2 | sylan9bb 509 | 1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1784 |
This theorem is referenced by: bj-ru0 35100 |
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