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Mirrors > Home > ILE Home > Th. List > sb1 | GIF version |
Description: One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sb1 | ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 1686 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | 1 | simprbi 269 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∃wex 1421 [wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 |
This theorem depends on definitions: df-bi 115 df-sb 1686 |
This theorem is referenced by: sbh 1699 sbiedh 1710 sb4a 1722 sb4e 1726 sbcof2 1731 sb4 1753 sb4or 1754 spsbe 1763 sbidm 1772 sb5rf 1773 bj-sbimedh 10582 |
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