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Mirrors > Home > MPE Home > Th. List > sbctt | Structured version Visualization version GIF version |
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
sbctt | ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3714 | . . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | 1 | bibi1d 345 | . . . 4 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ 𝜑) ↔ ([𝐴 / 𝑥]𝜑 ↔ 𝜑))) |
3 | 2 | imbi2d 342 | . . 3 ⊢ (𝑦 = 𝐴 → ((Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) ↔ (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)))) |
4 | sbft 2235 | . . 3 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
5 | 3, 4 | vtoclg 3513 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))) |
6 | 5 | imp 407 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 Ⅎwnf 1769 [wsb 2044 ∈ wcel 2083 [wsbc 3711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-sbc 3712 |
This theorem is referenced by: sbcgf 3779 csbtt 3832 |
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