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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 3841 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]⊤ ↔ ⊤)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ⊤wtru 1537 ∈ wcel 2113 Vcvv 3491 [wsbc 3768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-sbc 3769 |
This theorem is referenced by: (None) |
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