Theorem equidqe 39494

Description: equid 2026 with existential quantifier without using ax-c5 39455 or ax-5 1924. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
equidqe	⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥

Proof of Theorem equidqe

Step	Hyp	Ref	Expression
1		ax6fromc10 39468	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥
2		ax7 2030	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥))
3	2	pm2.43i 52	. . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥)
4	3	con3i 154	. . 3 ⊢ (¬ 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥)
5	4	alimi 1825	. 2 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥)
6	1, 5	mto 199	1 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥

Syntax hints:  ¬ wn 3  ∀wal 1552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-c7 39457  ax-c10 39458

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1794

This theorem is referenced by:  axc5sp1  39495  equidq  39496