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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 2689 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
2 | eleq2 2690 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | |
3 | 1, 2 | sylan9bb 736 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-cleq 2615 df-clel 2618 |
This theorem is referenced by: trel 4759 pwnss 4830 epelg 5030 preleq 8514 oemapval 8580 cantnf 8590 wemapwe 8594 nnsdomel 8816 cldval 20827 isufil 21707 umgr2v2enb1 26422 issiga 30174 matunitlindf 33407 wepwsolem 37612 aomclem8 37631 nelbr 41291 |
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