Theorem ax12wdemo 2141

Description: Example of an application of ax12w 2139 that results in an instance of ax-12 2185 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 2038 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 2121	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦))
2		elequ2 2129	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑤))
3	2	cbvalvw 2038	. . . 4 ⊢ (∀𝑥 𝑧 ∈ 𝑥 ↔ ∀𝑤 𝑧 ∈ 𝑤)
4	3	a1i 11	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑥 ↔ ∀𝑤 𝑧 ∈ 𝑤))
5		elequ1 2121	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ 𝑥 ↔ 𝑣 ∈ 𝑥))
6	5	albidv 1922	. . . . 5 ⊢ (𝑦 = 𝑣 → (∀𝑧 𝑦 ∈ 𝑥 ↔ ∀𝑧 𝑣 ∈ 𝑥))
7	6	cbvalvw 2038	. . . 4 ⊢ (∀𝑦∀𝑧 𝑦 ∈ 𝑥 ↔ ∀𝑣∀𝑧 𝑣 ∈ 𝑥)
8		elequ2 2129	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝑥 ↔ 𝑣 ∈ 𝑦))
9	8	albidv 1922	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑧 𝑣 ∈ 𝑥 ↔ ∀𝑧 𝑣 ∈ 𝑦))
10	9	albidv 1922	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑣∀𝑧 𝑣 ∈ 𝑥 ↔ ∀𝑣∀𝑧 𝑣 ∈ 𝑦))
11	7, 10	bitrid 283	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦∀𝑧 𝑦 ∈ 𝑥 ↔ ∀𝑣∀𝑧 𝑣 ∈ 𝑦))
12	1, 4, 11	3anbi123d 1439	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑦 ∧ ∀𝑤 𝑧 ∈ 𝑤 ∧ ∀𝑣∀𝑧 𝑣 ∈ 𝑦)))
13		elequ2 2129	. . 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 1441	. 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 2139	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 206   ∧ w3a 1087  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ex 1782

This theorem is referenced by: (None)