![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3791 | . 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 209 Ⅎwnf 1785 ∈ wcel 2111 Vcvv 3441 [wsbc 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-sbc 3721 |
This theorem is referenced by: csbgfi 3848 bnj110 32240 bnj1039 32353 mptsnunlem 34755 sbali 35550 sbexi 35551 |
Copyright terms: Public domain | W3C validator |