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Mirrors > Home > NFE Home > Th. List > elequ2 | GIF version |
Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
elequ2 | ⊢ (x = y → (z ∈ x ↔ z ∈ y)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-14 1714 | . 2 ⊢ (x = y → (z ∈ x → z ∈ y)) | |
2 | ax-14 1714 | . . 3 ⊢ (y = x → (z ∈ y → z ∈ x)) | |
3 | 2 | equcoms 1681 | . 2 ⊢ (x = y → (z ∈ y → z ∈ x)) |
4 | 1, 3 | impbid 183 | 1 ⊢ (x = y → (z ∈ x ↔ z ∈ y)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-14 1714 |
This theorem depends on definitions: df-bi 177 df-ex 1542 |
This theorem is referenced by: ax11wdemo 1723 dveel2 2020 elsb2 2104 dveel2ALT 2191 ax11el 2194 axext3 2336 axext4 2337 bm1.1 2338 ssfin 4471 ncfinlower 4484 nnadjoinlem1 4520 fv3 5342 |
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