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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 2137 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | |
2 | ax-13 2137 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧)) | |
3 | 2 | equcoms 1695 | . 2 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧)) |
4 | 1, 3 | impbid 128 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-gen 1436 ax-ie2 1481 ax-8 1491 ax-17 1513 ax-i9 1517 ax-13 2137 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: cleljust 2141 elsb3 2142 dveel1 2144 nalset 4106 zfpow 4148 mss 4198 zfun 4406 ctssdc 7069 acfun 7154 ccfunen 7196 bj-nalset 13612 bj-nnelirr 13670 nninfsellemqall 13729 nninfomni 13733 |
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