Theorem ax9o 1570

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 1568	. 2 ⊢ ∃x x = y
2		19.29r 1494	. . 3 ⊢ ((∃x x = y ∧ ∀x(x = y → ∀xφ)) → ∃x(x = y ∧ (x = y → ∀xφ)))
3		hba1 1415	. . . . 5 ⊢ (∀xφ → ∀x∀xφ)
4		pm3.35 329	. . . . 5 ⊢ ((x = y ∧ (x = y → ∀xφ)) → ∀xφ)
5	3, 4	exlimih 1466	. . . 4 ⊢ (∃x(x = y ∧ (x = y → ∀xφ)) → ∀xφ)
6		ax-4 1381	. . . 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 402	1 ⊢ (∀x(x = y → ∀xφ) → φ)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1226   = wceq 1228  ∃wex 1362

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-i9 1404  ax-ial 1409

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  equsalh  1596  spimth  1605  spimh  1607