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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 414 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝐵 = 𝐶)) |
4 | 1, 3 | alrimi 2207 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)) |
5 | csbiedf.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | csbiedf.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐶) | |
7 | csbiebt 3886 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | |
8 | 5, 6, 7 | syl2anc 585 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
9 | 4, 8 | mpbid 231 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 ⦋csb 3856 |
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-3an 1090 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3446 df-sbc 3741 df-csb 3857 |
This theorem is referenced by: csbiedOLD 3895 csbie2t 3897 fvmptdf 6955 fsumsplit1 15635 fprodsplit1f 15878 natpropd 17870 fucpropd 17871 gsummptf1o 19745 gsummpt2d 31940 mnringvald 42576 sumsnd 43319 |
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