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Mirrors > Home > MPE Home > Th. List > sbcel1g | Structured version Visualization version GIF version |
Description: Move proper substitution in and out of a membership relation. Note that the scope of [𝐴 / 𝑥] is the wff 𝐵 ∈ 𝐶, whereas the scope of ⦋𝐴 / 𝑥⦌ is the class 𝐵. (Contributed by NM, 10-Nov-2005.) |
Ref | Expression |
---|---|
sbcel1g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcel12 4417 | . 2 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | |
2 | csbconstg 3927 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐶) | |
3 | 2 | eleq2d 2825 | . 2 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝐶)) |
4 | 1, 3 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 [wsbc 3791 ⦋csb 3908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-nul 4340 |
This theorem is referenced by: rspcsbela 4444 csbopg 4896 fprodcllemf 15991 wunnat 18011 wunnatOLD 18012 catcfuccl 18173 catcfucclOLD 18174 esumpfinvalf 34057 esum2dlem 34073 measiuns 34198 bj-sbel1 36888 csbfinxpg 37371 finixpnum 37592 renegclALT 38945 cdlemk35s 40920 ellimcabssub0 45573 |
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