Theorem ax5eq 38404

Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-5 1906 considered as a metatheorem. Do not use it for later proofs - use ax-5 1906 instead, to avoid reference to the redundant axiom ax-c16 38364.) (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax5eq	⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem ax5eq

Step	Hyp	Ref	Expression
1		ax-c9 38362	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		ax-c16 38364	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
3		ax-c16 38364	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
4	1, 2, 3	pm2.61ii 183	1 ⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)

Syntax hints:   → wi 4  ∀wal 1532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-c9 38362  ax-c16 38364

This theorem is referenced by:  dveeq1-o16  38408