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Theorem stdpc4ALT 2590
 Description: Alternate version of stdpc4 2073. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dfsb1.ph (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Assertion
Ref Expression
stdpc4ALT (∀𝑥𝜑𝜃)

Proof of Theorem stdpc4ALT
StepHypRef Expression
1 ala1 1814 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
2 dfsb1.ph . . 3 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
32sb2ALT 2587 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → 𝜃)
41, 3syl 17 1 (∀𝑥𝜑𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177  ax-13 2390 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781 This theorem is referenced by:  sbftALT  2593
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