Theorem sb3 1841

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

Assertion

Ref	Expression
sb3	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))

Proof of Theorem sb3

Step	Hyp	Ref	Expression
1		equs5 1839	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		sb2 1777	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
3	1, 2	syl6 33	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1361  ∃wex 1502  [wsb 1772

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-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773

This theorem is referenced by: (None)