Theorem hbequid 33212

Description: Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof does not use ax-c10 33189.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbequid	⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)

Proof of Theorem hbequid

Step	Hyp	Ref	Expression
1		ax-c9 33193	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)))
2		ax7 1930	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥))
3	2	pm2.43i 50	. . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥)
4	3	alimi 1730	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
5	4	a1d 25	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
6	1, 5, 5	pm2.61ii 176	1 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)

Syntax hints:   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-c9 33193

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  nfequid-o  33213  equidq  33227