Theorem sb5 1764

Description: Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)

Assertion

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

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem sb5

Step	Hyp	Ref	Expression
1		sb6 1763	. 2 ⊢ ([y / x]φ ↔ ∀x(x = y → φ))
2		sb56 1762	. 2 ⊢ (∃x(x = y ∧ φ) ↔ ∀x(x = y → φ))
3	1, 2	bitr4i 176	1 ⊢ ([y / x]φ ↔ ∃x(x = y ∧ φ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-sb 1643

This theorem is referenced by:  sbnv  1765  sborv  1767  sbi2v  1769  nfsbxy  1815  nfsbxyt  1816  2sb5  1856  dfsb7  1864  sb7f  1865  sbexyz  1876  sbc5  2781