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Mirrors > Home > ILE Home > Th. List > csbied | GIF version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
csbied.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
csbied.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbied | ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1515 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfcvd 2307 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐶) | |
3 | csbied.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | csbied.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) | |
5 | 1, 2, 3, 4 | csbiedf 3081 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ⦋csb 3041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-sbc 2948 df-csb 3042 |
This theorem is referenced by: csbied2 3088 fvmptd 5562 seq3f1olemp 10428 fsumgcl 11317 fsum3 11318 fsumshftm 11376 fisum0diag2 11378 fprodseq 11514 fprodeq0 11548 |
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