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Theorem sbftALT 2593
Description: Alternate version of sbft 2271. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dfsb1.ph (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Assertion
Ref Expression
sbftALT (Ⅎ𝑥𝜑 → (𝜃𝜑))

Proof of Theorem sbftALT
StepHypRef Expression
1 dfsb1.ph . . . 4 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
21spsbeALT 2589 . . 3 (𝜃 → ∃𝑥𝜑)
3 19.9t 2205 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
42, 3syl5ib 247 . 2 (Ⅎ𝑥𝜑 → (𝜃𝜑))
5 nf5r 2194 . . 3 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
61stdpc4ALT 2590 . . 3 (∀𝑥𝜑𝜃)
75, 6syl6 35 . 2 (Ⅎ𝑥𝜑 → (𝜑𝜃))
84, 7impbid 215 1 (Ⅎ𝑥𝜑 → (𝜃𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  wnf 1785
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 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by:  sbfALT  2594
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