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Mirrors > Home > MPE Home > Th. List > csbiedf | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
csbiedf.1 | ⊢ Ⅎ𝑥𝜑 |
csbiedf.2 | ⊢ (𝜑 → Ⅎ𝑥𝐶) |
csbiedf.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
csbiedf.4 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbiedf | ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbiedf.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | csbiedf.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) | |
3 | 2 | ex 416 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝐵 = 𝐶)) |
4 | 1, 3 | alrimi 2211 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)) |
5 | csbiedf.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | csbiedf.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐶) | |
7 | csbiebt 3857 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | |
8 | 5, 6, 7 | syl2anc 587 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
9 | 4, 8 | mpbid 235 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ⦋csb 3828 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-sbc 3721 df-csb 3829 |
This theorem is referenced by: csbied 3864 csbie2t 3866 fvmptdf 6751 fprodsplit1f 15336 natpropd 17238 fucpropd 17239 gsummptf1o 19076 gsummpt2d 30734 mnringvald 40921 sumsnd 41655 fsumsplit1 42214 |
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