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Mirrors > Home > MPE Home > Th. List > Mathboxes > gm-sbtru | Structured version Visualization version GIF version |
Description: Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.) |
Ref | Expression |
---|---|
gm-sbtru.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
gm-sbtru | ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gm-sbtru.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sbcg 3774 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]⊤ ↔ ⊤)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ⊤wtru 1544 ∈ wcel 2110 Vcvv 3408 [wsbc 3694 |
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 2016 ax-8 2112 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-sb 2071 df-clab 2715 df-clel 2816 df-sbc 3695 |
This theorem is referenced by: (None) |
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