Theorem ax12wdemo 2176

Description: Example of an application of ax12w 2174 that results in an instance of ax-12 2219 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 2063 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 2156	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦))
2		elequ2 2164	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑤))
3	2	cbvalvw 2063	. . . 4 ⊢ (∀𝑥 𝑧 ∈ 𝑥 ↔ ∀𝑤 𝑧 ∈ 𝑤)
4	3	a1i 11	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑥 ↔ ∀𝑤 𝑧 ∈ 𝑤))
5		elequ1 2156	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ 𝑥 ↔ 𝑣 ∈ 𝑥))
6	5	albidv 1947	. . . . 5 ⊢ (𝑦 = 𝑣 → (∀𝑧 𝑦 ∈ 𝑥 ↔ ∀𝑧 𝑣 ∈ 𝑥))
7	6	cbvalvw 2063	. . . 4 ⊢ (∀𝑦∀𝑧 𝑦 ∈ 𝑥 ↔ ∀𝑣∀𝑧 𝑣 ∈ 𝑥)
8		elequ2 2164	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝑥 ↔ 𝑣 ∈ 𝑦))
9	8	albidv 1947	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑧 𝑣 ∈ 𝑥 ↔ ∀𝑧 𝑣 ∈ 𝑦))
10	9	albidv 1947	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑣∀𝑧 𝑣 ∈ 𝑥 ↔ ∀𝑣∀𝑧 𝑣 ∈ 𝑦))
11	7, 10	bitrid 286	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦∀𝑧 𝑦 ∈ 𝑥 ↔ ∀𝑣∀𝑧 𝑣 ∈ 𝑦))
12	1, 4, 11	3anbi123d 1462	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑦 ∧ ∀𝑤 𝑧 ∈ 𝑤 ∧ ∀𝑣∀𝑧 𝑣 ∈ 𝑦)))
13		elequ2 2164	. . 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 1464	. 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 2174	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 209   ∧ w3a 1101  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-ex 1807

This theorem is referenced by: (None)