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Mirrors > Home > MPE Home > Th. List > sbequ2ALT | Structured version Visualization version GIF version |
Description: Alternate version of sbequ2 2250. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1.ph | ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Ref | Expression |
---|---|
sbequ2ALT | ⊢ (𝑥 = 𝑦 → (𝜃 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsb1.ph | . . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | 1 | simplbi 500 | . 2 ⊢ (𝜃 → (𝑥 = 𝑦 → 𝜑)) |
3 | 2 | com12 32 | 1 ⊢ (𝑥 = 𝑦 → (𝜃 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 209 df-an 399 |
This theorem is referenced by: sbequ12ALT 2581 dfsb2ALT 2591 sbequiALT 2596 sbi1ALT 2606 |
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