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Mirrors > Home > MPE Home > Th. List > csbied2 | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
csbied2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
csbied2.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
csbied2.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
csbied2 | ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbied2.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
3 | csbied2.2 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 2, 3 | sylan9eqr 2881 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵) |
5 | csbied2.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷) | |
6 | 4, 5 | syldan 593 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷) |
7 | 1, 6 | csbied 3922 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⦋csb 3886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-sbc 3776 df-csb 3887 |
This theorem is referenced by: prdsval 16731 cidfval 16950 monfval 17005 idfuval 17149 isnat 17220 fucco 17235 catcval 17359 xpcval 17430 1stfval 17444 2ndfval 17447 prfval 17452 evlf2 17471 curfval 17476 hofval 17505 ipoval 17767 poimirlem2 34898 rngcvalALTV 44239 ringcvalALTV 44285 |
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