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Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1591.) 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 1652 | . 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 → 𝜑)) | |
2 | 1 | equcoms 1594 | 1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 [wsb 1645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-gen 1338 ax-ie2 1383 ax-8 1395 ax-17 1419 ax-i9 1423 |
This theorem depends on definitions: df-bi 110 df-sb 1646 |
This theorem is referenced by: ax16 1694 sbequi 1720 sb5rf 1732 |
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