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Mirrors > Home > MPE Home > Th. List > stdpc4ALT | Structured version Visualization version GIF version |
Description: Alternate version of stdpc4 2072. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1.ph | ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Ref | Expression |
---|---|
stdpc4ALT | ⊢ (∀𝑥𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ala1 1813 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | dfsb1.ph | . . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
3 | 2 | sb2ALT 2586 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜃) |
4 | 1, 3 | syl 17 | 1 ⊢ (∀𝑥𝜑 → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 ∃wex 1779 |
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-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 |
This theorem is referenced by: sbftALT 2592 |
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