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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 2964 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
2 | elex 2749 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ V) | |
3 | 1, 2 | sylbi 121 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 {cab 2163 Vcvv 2738 [wsbc 2963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-v 2740 df-sbc 2964 |
This theorem is referenced by: sbcco 2985 sbc5 2987 sbcan 3006 sbcor 3008 sbcn1 3011 sbcim1 3012 sbcbi1 3013 sbcal 3015 sbcex2 3017 sbcel1v 3026 sbcel21v 3028 sbcimdv 3029 sbcrext 3041 spesbc 3049 csbprc 3469 opelopabsb 4261 |
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