Theorem equidq 34209

Description: equid 1939 with universal quantifier without using ax-c5 34168 or ax-5 1839. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
equidq	⊢ ∀𝑦 𝑥 = 𝑥

Proof of Theorem equidq

Step	Hyp	Ref	Expression
1		equidqe 34207	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2		ax10fromc7 34180	. . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥)
3		hbequid 34194	. . . 4 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
4	3	con3i 150	. . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
5	2, 4	alrimih 1751	. 2 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥)
6	1, 5	mt3 192	1 ⊢ ∀𝑦 𝑥 = 𝑥

Syntax hints:  ¬ wn 3  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c9 34175

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)