Theorem sb5 1270

Description: Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40.

Assertion

Ref	Expression
sb5	⊢ ([y / x]φ ↔ ∃x(x = y ⋀ φ))

Distinct variable group:   x,y

Proof of Theorem sb5

Step	Hyp	Ref	Expression
1		sb6 1269	. 2 ⊢ ([y / x]φ ↔ ∀x(x = y → φ))
2		sb56 1268	. 2 ⊢ (∃x(x = y ⋀ φ) ↔ ∀x(x = y → φ))
3	1, 2	bitr4 176	1 ⊢ ([y / x]φ ↔ ∃x(x = y ⋀ φ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   = wceq 958  ∃wex 982  [wsbc 1172

This theorem is referenced by:  2sb5 1337  dfsb7 1342  sbelx 1346

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-16 1212  ax-11o 1220

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174

Copyright terms: Public domain