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Mirrors > Home > MPE Home > Th. List > eleq12 | Structured version Visualization version GIF version |
Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.) |
Ref | Expression |
---|---|
eleq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2676 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
2 | eleq2 2677 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | |
3 | 1, 2 | sylan9bb 732 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-cleq 2603 df-clel 2606 |
This theorem is referenced by: trel 4687 pwnss 4756 epelg 4950 preleq 8397 oemapval 8463 cantnf 8473 wemapwe 8477 nnsdomel 8699 cldval 20637 isufil 21517 issiga 29501 matunitlindf 32577 wepwsolem 36630 aomclem8 36649 umgr2v2enb1 40742 |
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