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Mirrors > Home > MPE Home > Th. List > sb4e | Structured version Visualization version GIF version |
Description: One direction of a simplified definition of substitution that unlike sb4b 2466 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2363. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb4e | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb1 2469 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
2 | equs5e 2449 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) | |
3 | 1, 2 | syl 17 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1531 ∃wex 1773 [wsb 2059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-12 2163 ax-13 2363 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-nf 1778 df-sb 2060 |
This theorem is referenced by: hbsb2e 2477 |
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