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Theorem sb6fALT 2604
Description: Alternate version of sb6f 2539. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dfsb1.p5 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
sb6fALT.1 𝑦𝜑
Assertion
Ref Expression
sb6fALT (𝜃 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb6fALT
StepHypRef Expression
1 dfsb1.p5 . . . 4 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
2 biid 264 . . . 4 (((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)) ↔ ((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)))
3 sb6fALT.1 . . . . 5 𝑦𝜑
43nf5ri 2197 . . . 4 (𝜑 → ∀𝑦𝜑)
51, 2, 4sbimiALT 2579 . . 3 (𝜃 → ((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)))
62sb4aALT 2600 . . 3 (((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)) → ∀𝑥(𝑥 = 𝑦𝜑))
75, 6syl 17 . 2 (𝜃 → ∀𝑥(𝑥 = 𝑦𝜑))
81sb2ALT 2589 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → 𝜃)
97, 8impbii 212 1 (𝜃 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2179  ax-13 2392
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:  sb5fALT  2605
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