Theorem sbtvOLD 2518

Description: Obsolete version of sbt 2070 as of 6-Jul-2023. A substitution into a theorem yields a theorem. See sbt 2070 when 𝑥, 𝑦 need not be disjoint. (Contributed by BJ, 31-May-2019.) Reduce axioms. (Revised by Steven Nguyen, 25-Apr-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
sbtvOLD.1	⊢ 𝜑

Assertion

Ref	Expression
sbtvOLD	⊢ [𝑥 / 𝑦]𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbtvOLD

Step	Hyp	Ref	Expression
1		sbtvOLD.1	. . 3 ⊢ 𝜑
2	1	a1i 11	. 2 ⊢ (𝑦 = 𝑥 → 𝜑)
3		ax6ev 1971	. . 3 ⊢ ∃𝑦 𝑦 = 𝑥
4	3, 1	exan 1861	. 2 ⊢ ∃𝑦(𝑦 = 𝑥 ∧ 𝜑)
5		dfsb1 2509	. 2 ⊢ ([𝑥 / 𝑦]𝜑 ↔ ((𝑦 = 𝑥 → 𝜑) ∧ ∃𝑦(𝑦 = 𝑥 ∧ 𝜑)))
6	2, 4, 5	mpbir2an 709	1 ⊢ [𝑥 / 𝑦]𝜑

Syntax hints:   → wi 4   ∧ wa 398  ∃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)