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Mirrors > Home > MPE Home > Th. List > 3eltr4g | Structured version Visualization version GIF version |
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.) |
Ref | Expression |
---|---|
3eltr4g.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
3eltr4g.2 | ⊢ 𝐶 = 𝐴 |
3eltr4g.3 | ⊢ 𝐷 = 𝐵 |
Ref | Expression |
---|---|
3eltr4g | ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eltr4g.2 | . . 3 ⊢ 𝐶 = 𝐴 | |
2 | 3eltr4g.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | 1, 2 | eqeltrid 2837 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
4 | 3eltr4g.3 | . 2 ⊢ 𝐷 = 𝐵 | |
5 | 3, 4 | eleqtrrdi 2844 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-cleq 2724 df-clel 2810 |
This theorem is referenced by: riotacl2 7381 rankelun 9866 rankelpr 9867 rankelop 9868 cdivcncf 24436 rrx0el 24914 itg1addlem4 25215 itg1addlem4OLD 25216 cxpcn3 26253 bposlem4 26787 nosepdm 27184 mirauto 27932 ldgenpisyslem1 33156 relowlpssretop 36240 0prjspnlem 41366 mapfzcons 41444 fourierdlem62 44874 fourierdlem63 44875 line2x 47430 line2y 47431 |
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