Theorem sbequ2 1869

Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)

Assertion

Ref	Expression
sbequ2	⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))

Proof of Theorem sbequ2

Step	Hyp	Ref	Expression
1		df-sb 1868	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	simplbi 475	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦 → 𝜑))
3	2	com12 32	1 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-an 385  df-sb 1868

This theorem is referenced by:  stdpc7  1945  sbequ12  2097  dfsb2  2361  sbequi  2363  sbi1  2380  bj-mo3OLD  32022  2pm13.193  37789  2pm13.193VD  38161