Theorem ax9o 1644

Description: An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Assertion

Ref	Expression
ax9o	⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Proof of Theorem ax9o

Step	Hyp	Ref	Expression
1		a9e 1642	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		19.29r 1568	. . 3 ⊢ ((∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) → ∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)))
3		hba1 1488	. . . . 5 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
4		pm3.35 342	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑)
5	3, 4	exlimih 1540	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑)
6		ax-4 1455	. . . 4 ⊢ (∀𝑥𝜑 → 𝜑)
7	5, 6	syl 14	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → 𝜑)
8	2, 7	syl 14	. 2 ⊢ ((∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) → 𝜑)
9	1, 8	mpan 418	1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1297   = wceq 1299  ∃wex 1436

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-i9 1478  ax-ial 1482

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  equsalh  1672  spimth  1681  spimh  1683