Theorem sb4OLD 2460

Description: Obsolete as of 30-Jul-2023. Use sb4b 2457 instead. One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 14-May-1993.) Revise df-sb 2043. (Revised by Wolf Lammen, 25-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sb4OLD	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Proof of Theorem sb4OLD

Step	Hyp	Ref	Expression
1		sb4b 2457	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2	1	biimpd 230	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1520  [wsb 2042

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  df-sb 2043

This theorem is referenced by:  sbequiOLD  2488  sbi1OLD  2496