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Mirrors > Home > MPE Home > Th. List > csbief | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
csbief.1 | ⊢ 𝐴 ∈ V |
csbief.2 | ⊢ Ⅎ𝑥𝐶 |
csbief.3 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbief | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbief.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | csbief.2 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶) |
4 | csbief.3 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
5 | 3, 4 | csbiegf 3523 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 Vcvv 3173 ⦋csb 3499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-csb 3500 |
This theorem is referenced by: csbie 3525 csbun 3961 csbin 3962 csbif 4088 csbopab 4932 csbopabgALT 4933 csbima12 5402 csbiota 5797 csbriota 6523 csbov123 6585 pcmpt 15434 mpfrcl 19339 iundisj2 23124 iundisj2f 28785 iundisj2fi 28943 csbdif 32347 sbccom2f 33101 csbcog 36960 csbingOLD 38076 csbafv12g 39866 csbaovg 39909 |
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