Theorem sb2 1815

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 1559	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
2		equs4 1773	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
3		df-sb 1811	. 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 1396  ∃wex 1541  [wsb 1810

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-i9 1579  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-sb 1811

This theorem is referenced by:  stdpc4  1823  equsb1  1833  equsb2  1834  sbiedh  1835  sb6f  1851  hbsb2a  1854  hbsb2e  1855  sbcof2  1858  sb3  1879  sb4b  1882  sb4bor  1883  hbsb2  1884  nfsb2or  1885  sb6rf  1901  sbi1v  1940  sbalyz  2052  iota4  5313