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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 2765 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
2 | elex 2566 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ V) | |
3 | 1, 2 | sylbi 114 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 {cab 2026 Vcvv 2557 [wsbc 2764 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 df-sbc 2765 |
This theorem is referenced by: sbcco 2785 sbc5 2787 sbcan 2805 sbcor 2807 sbcal 2810 sbcex2 2812 sbcrext 2835 spesbc 2843 csbprc 3262 opelopabsb 3997 |
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