Theorem equidq 33227

Description: equid 1926 with universal quantifier without using ax-c5 33186 or ax-5 1827. (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 33225	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2		ax10fromc7 33198	. . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥)
3		hbequid 33212	. . . 4 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
4	3	con3i 149	. . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
5	2, 4	alrimih 1741	. 2 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥)
6	1, 5	mt3 191	1 ⊢ ∀𝑦 𝑥 = 𝑥

Syntax hints:  ¬ wn 3  ∀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-c5 33186  ax-c4 33187  ax-c7 33188  ax-c10 33189  ax-c9 33193

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

This theorem is referenced by: (None)