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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 disjoint variable condition (sb6 2081) or a nonfreeness hypothesis (sb6f 2491). Usage of this theorem is discouraged because it depends on ax-13 2366. (Contributed by NM, 13-May-1993.) Revise df-sb 2061. (Revised by Wolf Lammen, 26-Jul-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb2 | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.27 42 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝑦 → 𝜑) → 𝜑)) | |
2 | 1 | al2imi 1810 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥𝜑)) |
3 | stdpc4 2064 | . . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | syl6 35 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)) |
5 | sb4b 2469 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
6 | 5 | biimprd 247 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)) |
7 | 4, 6 | pm2.61i 182 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1532 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2164 ax-13 2366 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: hbsb2 2476 hbsb2a 2478 hbsb2e 2480 equsb1 2485 equsb2 2486 dfsb2 2487 sb6f 2491 sbeqal1 43807 |
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