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Mirrors > Home > ILE Home > Th. List > elequ1 | GIF version |
Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
elequ1 | ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-13 2150 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | |
2 | ax-13 2150 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧)) | |
3 | 2 | equcoms 1708 | . 2 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧)) |
4 | 1, 3 | impbid 129 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-gen 1449 ax-ie2 1494 ax-8 1504 ax-17 1526 ax-i9 1530 ax-13 2150 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: cleljust 2154 elsb1 2155 dveel1 2157 nalset 4132 zfpow 4174 mss 4225 zfun 4433 ctssdc 7109 acfun 7203 ccfunen 7260 bj-nalset 14507 bj-nnelirr 14565 nninfsellemqall 14624 nninfomni 14628 |
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