Theorem equsb3 2102

Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)

Assertion

Ref	Expression
equsb3	⊢ ([y / x]x = z ↔ y = z)

Distinct variable group:   x,z

Proof of Theorem equsb3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equsb3lem 2101	. . 3 ⊢ ([w / x]x = z ↔ w = z)
2	1	sbbii 1653	. 2 ⊢ ([y / w][w / x]x = z ↔ [y / w]w = z)
3		nfv 1619	. . 3 ⊢ Ⅎw x = z
4	3	sbco2 2086	. 2 ⊢ ([y / w][w / x]x = z ↔ [y / x]x = z)
5		equsb3lem 2101	. 2 ⊢ ([y / w]w = z ↔ y = z)
6	2, 4, 5	3bitr3i 266	1 ⊢ ([y / x]x = z ↔ y = z)

Syntax hints:   ↔ wb 176  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sb8eu  2222  sb8iota  4347