Theorem elsb1 2182

Description: Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2183 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 1548	. . . . 5 ⊢ (𝑥 ∈ 𝑧 → ∀𝑤 𝑥 ∈ 𝑧)
2		elequ1 2179	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
3	1, 2	sbieh 1812	. . . 4 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧)
4	3	sbbii 1787	. . 3 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝑧 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝑧)
5		ax-17 1548	. . . 4 ⊢ (𝑤 ∈ 𝑧 → ∀𝑥 𝑤 ∈ 𝑧)
6	5	sbco2h 1991	. . 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 1807	. . . 4 ⊢ [𝑦 / 𝑤]𝑤 = 𝑦
9		elequ1 2179	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
10	9	sbimi 1786	. . . 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 1986	. . 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 1548	. . 3 ⊢ (𝑦 ∈ 𝑧 → ∀𝑤 𝑦 ∈ 𝑧)
15	14	sbh 1798	. 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 1784

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785

This theorem is referenced by:  cvjust  2199