![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4409 | . 2 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | |
2 | csbconstg 3913 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐶) | |
3 | 2 | eleq2d 2820 | . 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 205 ∈ wcel 2107 [wsbc 3778 ⦋csb 3894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-nul 4324 |
This theorem is referenced by: rspcsbela 4436 csbopg 4892 fprodcllemf 15902 wunnat 17907 wunnatOLD 17908 catcfuccl 18069 catcfucclOLD 18070 esumpfinvalf 33074 esum2dlem 33090 measiuns 33215 bj-sbel1 35785 csbfinxpg 36269 finixpnum 36473 renegclALT 37833 cdlemk35s 39808 ellimcabssub0 44333 |
Copyright terms: Public domain | W3C validator |