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Theorem sb5fALT 2557
 Description: Alternate version of sb5f 2492. (Contributed by NM, 5-Aug-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
sb5fALT (𝜃 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb5fALT
StepHypRef Expression
1 dfsb1.p5 . . 3 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
2 sb6fALT.1 . . 3 𝑦𝜑
31, 2sb6fALT 2556 . 2 (𝜃 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
42equs45f 2439 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
53, 4bitr4i 279 1 (𝜃 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520  ∃wex 1761  Ⅎwnf 1765 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766 This theorem is referenced by:  sb7fALT  2570
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