Theorem alcomiw 1704

Description: Weak version of alcom 1737. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)

Hypothesis

Ref	Expression
alcomiw.1	⊢ (y = z → (φ ↔ ψ))

Assertion

Ref	Expression
alcomiw	⊢ (∀x∀yφ → ∀y∀xφ)

Distinct variable groups:   y,z   x,y   φ,z   ψ,y

Allowed substitution hints:   φ(x,y)   ψ(x,z)

Proof of Theorem alcomiw

Step	Hyp	Ref	Expression
1		alcomiw.1	. . . . 5 ⊢ (y = z → (φ ↔ ψ))
2	1	biimpd 198	. . . 4 ⊢ (y = z → (φ → ψ))
3	2	cbvalivw 1674	. . 3 ⊢ (∀yφ → ∀zψ)
4	3	alimi 1559	. 2 ⊢ (∀x∀yφ → ∀x∀zψ)
5		ax-17 1616	. 2 ⊢ (∀x∀zψ → ∀y∀x∀zψ)
6	1	biimprd 214	. . . . . 6 ⊢ (y = z → (ψ → φ))
7	6	equcoms 1681	. . . . 5 ⊢ (z = y → (ψ → φ))
8	7	spimvw 1669	. . . 4 ⊢ (∀zψ → φ)
9	8	alimi 1559	. . 3 ⊢ (∀x∀zψ → ∀xφ)
10	9	alimi 1559	. 2 ⊢ (∀y∀x∀zψ → ∀y∀xφ)
11	4, 5, 10	3syl 18	1 ⊢ (∀x∀yφ → ∀y∀xφ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  hbalw  1709  ax7w  1718