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Mirrors > Home > MPE Home > Th. List > sbhypf | Structured version Visualization version GIF version |
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3908. (Contributed by Raph Levien, 10-Apr-2004.) |
Ref | Expression |
---|---|
sbhypf.1 | ⊢ Ⅎ𝑥𝜓 |
sbhypf.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbhypf | ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2822 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
2 | 1 | equsexvw 2002 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝐴) ↔ 𝑦 = 𝐴) |
3 | nfs1v 2264 | . . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
4 | sbhypf.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfbi 1895 | . . 3 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | sbequ12 2243 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
7 | 6 | bicomd 224 | . . . 4 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) |
8 | sbhypf.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
9 | 7, 8 | sylan9bb 510 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
10 | 5, 9 | exlimi 2207 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
11 | 2, 10 | sylbir 236 | 1 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 Ⅎwnf 1775 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-10 2136 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-cleq 2811 |
This theorem is referenced by: mob2 3703 reu2eqd 3724 cbvrabcsfw 3921 cbvopab1 5130 ralxpf 5710 cbviotaw 6314 cbvriotaw 7112 tfisi 7562 ac6sf 9899 nn0ind-raph 12070 ac6sf2 30298 nn0min 30463 ac6gf 34888 fdc1 34902 |
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