Theorem sb2 1816

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 1812	. 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 1811

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 1812

This theorem is referenced by:  stdpc4  1824  equsb1  1834  equsb2  1835  sbiedh  1836  sb6f  1852  hbsb2a  1855  hbsb2e  1856  sbcof2  1859  sb3  1880  sb4b  1883  sb4bor  1884  hbsb2  1885  nfsb2or  1886  sb6rf  1902  sbi1v  1941  sbalyz  2053  iota4  5332