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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 2793 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵) |
5 | csbied2.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷) | |
6 | 4, 5 | syldan 594 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷) |
7 | 1, 6 | csbied 3836 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⦋csb 3798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-sbc 3684 df-csb 3799 |
This theorem is referenced by: prdsval 16914 cidfval 17133 monfval 17191 idfuval 17336 isnat 17408 fucco 17425 catcval 17560 xpcval 17638 1stfval 17652 2ndfval 17655 prfval 17660 evlf2 17680 curfval 17685 hofval 17714 ipoval 17990 mntoval 30933 mgcoval 30937 poimirlem2 35465 rngcvalALTV 45135 ringcvalALTV 45181 |
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