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Mirrors > Home > MPE Home > Th. List > sb2ALT | Structured version Visualization version GIF version |
Description: Alternate version of sb2 2493. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1.ph | ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Ref | Expression |
---|---|
sb2ALT | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2180 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
2 | equs4 2427 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
3 | dfsb1.ph | . 2 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
4 | 1, 2, 3 | sylanbrc 586 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: stdpc4ALT 2566 dfsb2ALT 2567 sbequiALT 2572 hbsb2ALT 2575 equsb1ALT 2577 sb6fALT 2578 sbi1ALT 2582 |
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