Theorem sb2vOLD 2097

Description: Obsolete as of 30-Jul-2023. Use sb6 2093 instead. Version of sb2 2504 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by BJ, 31-May-2019.) Revise df-sb 2070. (Revised by Steven Nguyen, 8-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sb2vOLD	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb2vOLD

Step	Hyp	Ref	Expression
1		sb6 2093	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	biimpri 230	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1535  [wsb 2069

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

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070

This theorem is referenced by:  equsb1vOLD  2113  sb6OLD  2280  sbi1vOLD  2323