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Mirrors > Home > MPE Home > Th. List > stdpc7 | Structured version Visualization version GIF version |
Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1944.) 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, 𝑥)." 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 1869 | . 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 → 𝜑)) | |
2 | 1 | equcoms 1934 | 1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-sb 1868 |
This theorem is referenced by: (None) |
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