Theorem ax12wdemo 2014

Description: Example of an application of ax12w 2012 that results in an instance of ax-12 2049 for a contrived formula with mixed free and bound variables, (𝑥 ∈ 𝑦 ∧ ∀𝑥𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧𝑦 ∈ 𝑥), in place of 𝜑. The proof illustrates bound variable renaming with cbvalvw 1971 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.)

Assertion

Ref	Expression
ax12wdemo	⊢ (𝑥 = 𝑦 → (∀𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥) → ∀𝑥(𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥))))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem ax12wdemo

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ1 1999	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦))
2		elequ2 2006	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑤))
3	2	cbvalvw 1971	. . . 4 ⊢ (∀𝑥 𝑧 ∈ 𝑥 ↔ ∀𝑤 𝑧 ∈ 𝑤)
4	3	a1i 11	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑥 ↔ ∀𝑤 𝑧 ∈ 𝑤))
5		elequ1 1999	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ 𝑥 ↔ 𝑣 ∈ 𝑥))
6	5	albidv 1851	. . . . 5 ⊢ (𝑦 = 𝑣 → (∀𝑧 𝑦 ∈ 𝑥 ↔ ∀𝑧 𝑣 ∈ 𝑥))
7	6	cbvalvw 1971	. . . 4 ⊢ (∀𝑦∀𝑧 𝑦 ∈ 𝑥 ↔ ∀𝑣∀𝑧 𝑣 ∈ 𝑥)
8		elequ2 2006	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝑥 ↔ 𝑣 ∈ 𝑦))
9	8	albidv 1851	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑧 𝑣 ∈ 𝑥 ↔ ∀𝑧 𝑣 ∈ 𝑦))
10	9	albidv 1851	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑣∀𝑧 𝑣 ∈ 𝑥 ↔ ∀𝑣∀𝑧 𝑣 ∈ 𝑦))
11	7, 10	syl5bb 272	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦∀𝑧 𝑦 ∈ 𝑥 ↔ ∀𝑣∀𝑧 𝑣 ∈ 𝑦))
12	1, 4, 11	3anbi123d 1396	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑦 ∧ ∀𝑤 𝑧 ∈ 𝑤 ∧ ∀𝑣∀𝑧 𝑣 ∈ 𝑦)))
13		elequ2 2006	. . 3 ⊢ (𝑦 = 𝑣 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑣))
14	7	a1i 11	. . 3 ⊢ (𝑦 = 𝑣 → (∀𝑦∀𝑧 𝑦 ∈ 𝑥 ↔ ∀𝑣∀𝑧 𝑣 ∈ 𝑥))
15	13, 14	3anbi13d 1398	. 2 ⊢ (𝑦 = 𝑣 → ((𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑣 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑣∀𝑧 𝑣 ∈ 𝑥)))
16	12, 15	ax12w 2012	1 ⊢ (𝑥 = 𝑦 → (∀𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥) → ∀𝑥(𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1036  ∀wal 1478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1038  df-ex 1702

This theorem is referenced by: (None)