Theorem sb4vOLDOLD 2512

Description: Obsolete version of sb4vOLD 2095 as of 8-Jul-2023. Version of sb4OLD 2505 with a disjoint variable condition instead of a distinctor antecedent, which does not require ax-13 2389. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sb4vOLDOLD	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb4vOLDOLD

Step	Hyp	Ref	Expression
1		sb1 2502	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		sb56 2276	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	sylib 220	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534  ∃wex 1779  [wsb 2068

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-10 2144  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-sb 2069

This theorem is referenced by: (None)