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Mirrors > Home > MPE Home > Th. List > sb1 | Structured version Visualization version GIF version |
Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb5 2272) or a non-freeness hypothesis (sb5f 2532). (Contributed by NM, 13-May-1993.) |
Ref | Expression |
---|---|
sb1 | ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 2049 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | 1 | simprbi 478 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∃wex 1851 [wsb 2048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 197 df-an 383 df-sb 2049 |
This theorem is referenced by: spsbe 2052 sb6 2271 sb3b 2501 sb4 2502 sb4a 2503 sb4e 2508 sb6OLD 2575 bj-sb4v 33087 bj-sb6 33097 |
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