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Mirrors > Home > MPE Home > Th. List > sb5fALT | Structured version Visualization version GIF version |
Description: Alternate version of sb5f 2537. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1.p5 | ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
sb6fALT.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb5fALT | ⊢ (𝜃 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsb1.p5 | . . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | sb6fALT.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | 1, 2 | sb6fALT 2601 | . 2 ⊢ (𝜃 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
4 | 2 | equs45f 2481 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
5 | 3, 4 | bitr4i 280 | 1 ⊢ (𝜃 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 ∃wex 1779 Ⅎwnf 1783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 |
This theorem is referenced by: sb7fALT 2615 |
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