Theorem clelsb1 2455

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2103). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
clelsb1	⊢ ([y / x]x ∈ A ↔ y ∈ A)

Distinct variable group:   x,A

Allowed substitution hint:   A(y)

Proof of Theorem clelsb1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 ⊢ Ⅎx w ∈ A
2	1	sbco2 2086	. 2 ⊢ ([y / x][x / w]w ∈ A ↔ [y / w]w ∈ A)
3		nfv 1619	. . . 4 ⊢ Ⅎw x ∈ A
4		eleq1 2413	. . . 4 ⊢ (w = x → (w ∈ A ↔ x ∈ A))
5	3, 4	sbie 2038	. . 3 ⊢ ([x / w]w ∈ A ↔ x ∈ A)
6	5	sbbii 1653	. 2 ⊢ ([y / x][x / w]w ∈ A ↔ [y / x]x ∈ A)
7		nfv 1619	. . 3 ⊢ Ⅎw y ∈ A
8		eleq1 2413	. . 3 ⊢ (w = y → (w ∈ A ↔ y ∈ A))
9	7, 8	sbie 2038	. 2 ⊢ ([y / w]w ∈ A ↔ y ∈ A)
10	2, 6, 9	3bitr3i 266	1 ⊢ ([y / x]x ∈ A ↔ y ∈ A)

Syntax hints:   ↔ wb 176  [wsb 1648   ∈ wcel 1710

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  ax-ext 2334

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  df-cleq 2346  df-clel 2349

This theorem is referenced by:  hblem  2458  cbvreu  2834  sbcel1gv  3106  rmo3  3134