Theorem sb5 1267

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 1266	. 2 ⊢ ([y / x]φ ↔ ∀x(x = y → φ))
2		sb56 1265	. 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 953   = wceq 955  ∃wex 979  [wsbc 1169

This theorem is referenced by:  2sb5 1334  dfsb7 1339  sbelx 1343

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-16 1209  ax-11o 1217

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171

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