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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 1643 | . 2 ⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ))) | |
2 | 1 | simprbi 260 | 1 ⊢ ([y / x]φ → ∃x(x = y ∧ φ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∃wex 1378 [wsb 1642 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 |
This theorem depends on definitions: df-bi 110 df-sb 1643 |
This theorem is referenced by: sbh 1656 sbiedh 1667 sb4a 1679 sb4e 1683 sbcof2 1688 sb4 1710 sb4or 1711 spsbe 1720 sbidm 1728 sb5rf 1729 bj-sbimedh 9246 |
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