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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 2919 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
4 | 3eltr4g.3 | . 2 ⊢ 𝐷 = 𝐵 | |
5 | 3, 4 | eleqtrrdi 2926 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2816 df-clel 2895 |
This theorem is referenced by: riotacl2 7132 rankelun 9303 rankelpr 9304 rankelop 9305 cdivcncf 23527 rrx0el 24003 itg1addlem4 24302 cxpcn3 25331 bposlem4 25865 mirauto 26472 ldgenpisyslem1 31424 nosepdm 33190 relowlpssretop 34647 0prjspnlem 39275 mapfzcons 39320 fourierdlem62 42460 fourierdlem63 42461 line2x 44748 line2y 44749 |
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