Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 1491 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | |
2 | ax-13 1491 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧)) | |
3 | 2 | equcoms 1684 | . 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 1425 ax-ie2 1470 ax-8 1482 ax-13 1491 ax-17 1506 ax-i9 1510 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: cleljust 1910 elsb3 1951 dveel1 1995 nalset 4058 zfpow 4099 mss 4148 zfun 4356 ctssdc 6998 acfun 7063 ccfunen 7079 bj-nalset 13093 bj-nnelirr 13151 nninfsellemqall 13211 nninfomni 13215 |
Copyright terms: Public domain | W3C validator |