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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 7384 rankelun 9869 rankelpr 9870 rankelop 9871 cdivcncf 24661 rrx0el 25139 itg1addlem4 25440 itg1addlem4OLD 25441 cxpcn3 26480 bposlem4 27014 nosepdm 27411 mirauto 28190 ldgenpisyslem1 33447 relowlpssretop 36548 0prjspnlem 41667 mapfzcons 41756 fourierdlem62 45183 fourierdlem63 45184 line2x 47528 line2y 47529 |
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