Theorem elsb2 2104

Description: Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2103 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
elsb2	⊢ ([y / x]z ∈ x ↔ z ∈ y)

Distinct variable group:   x,z

Proof of Theorem elsb2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 ⊢ Ⅎx z ∈ w
2	1	sbco2 2086	. 2 ⊢ ([y / x][x / w]z ∈ w ↔ [y / w]z ∈ w)
3		nfv 1619	. . . 4 ⊢ Ⅎw z ∈ x
4		elequ2 1715	. . . 4 ⊢ (w = x → (z ∈ w ↔ z ∈ x))
5	3, 4	sbie 2038	. . 3 ⊢ ([x / w]z ∈ w ↔ z ∈ x)
6	5	sbbii 1653	. 2 ⊢ ([y / x][x / w]z ∈ w ↔ [y / x]z ∈ x)
7		nfv 1619	. . 3 ⊢ Ⅎw z ∈ y
8		elequ2 1715	. . 3 ⊢ (w = y → (z ∈ w ↔ z ∈ y))
9	7, 8	sbie 2038	. 2 ⊢ ([y / w]z ∈ w ↔ z ∈ y)
10	2, 6, 9	3bitr3i 266	1 ⊢ ([y / x]z ∈ x ↔ z ∈ y)

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-14 1714  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: (None)