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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbf3 | Structured version Visualization version GIF version | ||
| Description: Substitution has no effect on a bound variable (existential quantifier case); see sbf2 2272. (Contributed by BJ, 2-May-2019.) | 
| Ref | Expression | 
|---|---|
| bj-sbf3 | ⊢ ([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfe1 2150 | . 2 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
| 2 | 1 | sbf 2271 | 1 ⊢ ([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∃wex 1779 [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-10 2141 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: (None) | 
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