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Mirrors > Home > ILE Home > Th. List > stdpc7 | GIF version |
Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1680.) Translated to traditional notation, it can be read: "𝑥 = 𝑦 → (𝜑(𝑥, 𝑥) → 𝜑(𝑥, 𝑦)), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥, 𝑦)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.) |
Ref | Expression |
---|---|
stdpc7 | ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ2 1743 | . 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 → 𝜑)) | |
2 | 1 | equcoms 1685 | 1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 [wsb 1736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-gen 1426 ax-ie2 1471 ax-8 1483 ax-17 1507 ax-i9 1511 |
This theorem depends on definitions: df-bi 116 df-sb 1737 |
This theorem is referenced by: ax16 1786 sbequi 1812 sb5rf 1825 |
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