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Mirrors > Home > MPE Home > Th. List > sbcgfi | Structured version Visualization version GIF version |
Description: Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.) |
Ref | Expression |
---|---|
sbcgfi.1 | ⊢ 𝐴 ∈ V |
sbcgfi.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
sbcgfi | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcgfi.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sbcgfi.2 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | sbcgf 3868 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 Ⅎwnf 1780 ∈ wcel 2106 Vcvv 3478 [wsbc 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-sbc 3792 |
This theorem is referenced by: csbgfi 3929 bnj110 34851 bnj1039 34964 mptsnunlem 37321 sbali 38099 sbexi 38100 |
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