Theorem sb2 1781

Description: One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sb2	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)

Proof of Theorem sb2

Step	Hyp	Ref	Expression
1		ax-4 1524	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
2		equs4 1739	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
3		df-sb 1777	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
4	1, 2, 3	sylanbrc 417	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362  ∃wex 1506  [wsb 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by:  stdpc4  1789  equsb1  1799  equsb2  1800  sbiedh  1801  sb6f  1817  hbsb2a  1820  hbsb2e  1821  sbcof2  1824  sb3  1845  sb4b  1848  sb4bor  1849  hbsb2  1850  nfsb2or  1851  sb6rf  1867  sbi1v  1906  sbalyz  2018  iota4  5238