Theorem elsb1 2185

Description: Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2186 for substitution for the second argument. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
elsb1	⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)

Distinct variable group:   𝑥,𝑧

Proof of Theorem elsb1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1550	. . . . 5 ⊢ (𝑥 ∈ 𝑧 → ∀𝑤 𝑥 ∈ 𝑧)
2		elequ1 2182	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
3	1, 2	sbieh 1814	. . . 4 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧)
4	3	sbbii 1789	. . 3 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝑧 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝑧)
5		ax-17 1550	. . . 4 ⊢ (𝑤 ∈ 𝑧 → ∀𝑥 𝑤 ∈ 𝑧)
6	5	sbco2h 1993	. . 3 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝑧 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝑧)
7	4, 6	bitr3i 186	. 2 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑧 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝑧)
8		equsb1 1809	. . . 4 ⊢ [𝑦 / 𝑤]𝑤 = 𝑦
9		elequ1 2182	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
10	9	sbimi 1788	. . . 4 ⊢ ([𝑦 / 𝑤]𝑤 = 𝑦 → [𝑦 / 𝑤](𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
11	8, 10	ax-mp 5	. . 3 ⊢ [𝑦 / 𝑤](𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)
12		sbbi 1988	. . 3 ⊢ ([𝑦 / 𝑤](𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) ↔ ([𝑦 / 𝑤]𝑤 ∈ 𝑧 ↔ [𝑦 / 𝑤]𝑦 ∈ 𝑧))
13	11, 12	mpbi 145	. 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝑧 ↔ [𝑦 / 𝑤]𝑦 ∈ 𝑧)
14		ax-17 1550	. . 3 ⊢ (𝑦 ∈ 𝑧 → ∀𝑤 𝑦 ∈ 𝑧)
15	14	sbh 1800	. 2 ⊢ ([𝑦 / 𝑤]𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)
16	7, 13, 15	3bitri 206	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)

Syntax hints:   ↔ wb 105  [wsb 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  cvjust  2202