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Mirrors > Home > MPE Home > Th. List > sbequALT | Structured version Visualization version GIF version |
Description: Alternate version of sbequ 2088. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1.xz | ⊢ (𝜃 ↔ ((𝑧 = 𝑥 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑥 ∧ 𝜑))) |
dfsb1.yz | ⊢ (𝜏 ↔ ((𝑧 = 𝑦 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝜑))) |
Ref | Expression |
---|---|
sbequALT | ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsb1.xz | . . 3 ⊢ (𝜃 ↔ ((𝑧 = 𝑥 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑥 ∧ 𝜑))) | |
2 | dfsb1.yz | . . 3 ⊢ (𝜏 ↔ ((𝑧 = 𝑦 → 𝜑) ∧ ∃𝑧(𝑧 = 𝑦 ∧ 𝜑))) | |
3 | 1, 2 | sbequiALT 2572 | . 2 ⊢ (𝑥 = 𝑦 → (𝜃 → 𝜏)) |
4 | 2, 1 | sbequiALT 2572 | . . 3 ⊢ (𝑦 = 𝑥 → (𝜏 → 𝜃)) |
5 | 4 | equcoms 2027 | . 2 ⊢ (𝑥 = 𝑦 → (𝜏 → 𝜃)) |
6 | 3, 5 | impbid 215 | 1 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∃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-10 2142 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 |
This theorem is referenced by: sbco2ALT 2591 |
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