Theorem ax9o 1698

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 1696	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		19.29r 1621	. . 3 ⊢ ((∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) → ∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)))
3		hba1 1540	. . . . 5 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
4		pm3.35 347	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑)
5	3, 4	exlimih 1593	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑)
6		ax-4 1510	. . . 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 424	1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1351   = wceq 1353  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  equsalh  1726  spimth  1735  spimh  1737