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Mirrors > Home > ILE Home > Th. List > sbcex | GIF version |
Description: By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
sbcex | ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 2905 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
2 | elex 2692 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ V) | |
3 | 1, 2 | sylbi 120 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 {cab 2123 Vcvv 2681 [wsbc 2904 |
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-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-v 2683 df-sbc 2905 |
This theorem is referenced by: sbcco 2925 sbc5 2927 sbcan 2946 sbcor 2948 sbcn1 2951 sbcim1 2952 sbcbi1 2953 sbcal 2955 sbcex2 2957 sbcel1v 2966 sbcel21v 2968 sbcimdv 2969 sbcrext 2981 spesbc 2989 csbprc 3403 opelopabsb 4177 |
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