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Mirrors > Home > MPE Home > Th. List > hbsb2e | Structured version Visualization version GIF version |
Description: Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Ref | Expression |
---|---|
hbsb2e | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb4e 2445 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) | |
2 | sb2 2424 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) → [𝑦 / 𝑥]∃𝑦𝜑) | |
3 | 2 | axc4i 2262 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑) |
4 | 1, 3 | syl 17 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1505 ∃wex 1742 [wsb 2015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-10 2079 ax-12 2106 ax-13 2301 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-ex 1743 df-nf 1747 df-sb 2016 |
This theorem is referenced by: (None) |
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