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Mirrors > Home > MPE Home > Th. List > sb2 | Structured version Visualization version GIF version |
Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2417) or a non-freeness hypothesis (sb6f 2373). (Contributed by NM, 13-May-1993.) |
Ref | Expression |
---|---|
sb2 | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2041 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
2 | equs4 2278 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
3 | df-sb 1868 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
4 | 1, 2, 3 | sylanbrc 695 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 [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 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-sb 1868 |
This theorem is referenced by: stdpc4 2341 sb3 2343 sb4b 2346 hbsb2 2347 hbsb2a 2349 hbsb2e 2351 equsb1 2356 equsb2 2357 dfsb2 2361 sbequi 2363 sb6f 2373 sbi1 2380 sb6 2417 iota4 5786 wl-lem-moexsb 32529 sbeqal1 37620 |
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