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| Description: Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.) | 
| Ref | Expression | 
|---|---|
| sbft | ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | spsbe 2082 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑) | |
| 2 | 19.9t 2204 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
| 3 | 1, 2 | imbitrid 244 | . 2 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → 𝜑)) | 
| 4 | nf5r 2194 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
| 5 | stdpc4 2068 | . . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
| 6 | 4, 5 | syl6 35 | . 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → [𝑦 / 𝑥]𝜑)) | 
| 7 | 3, 6 | impbid 212 | 1 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 [wsb 2064 | 
| 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 1967 ax-7 2007 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 df-sb 2065 | 
| This theorem is referenced by: sbf 2271 sbctt 3860 wl-sbrimt 37548 wl-sblimt 37549 wl-sb8ft 37551 wl-equsb4 37558 ichnfimlem 47450 | 
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