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Mirrors > Home > MPE Home > Th. List > csbtt | Structured version Visualization version GIF version |
Description: Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
csbtt | ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 3909 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | nfcr 2893 | . . . 4 ⊢ (Ⅎ𝑥𝐵 → Ⅎ𝑥 𝑦 ∈ 𝐵) | |
3 | sbctt 3867 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥 𝑦 ∈ 𝐵) → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | |
4 | 2, 3 | sylan2 593 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) |
5 | 4 | eqabcdv 2874 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = 𝐵) |
6 | 1, 5 | eqtrid 2787 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 {cab 2712 Ⅎwnfc 2888 [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-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-sbc 3792 df-csb 3909 |
This theorem is referenced by: csbconstgf 3926 sbnfc2 4445 csbie2df 4449 constlimc 45580 |
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