Theorem ax7w 1718

Description: Weak version of ax-7 1734 from which we can prove any ax-7 1734 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-7 1734, this theorem requires that x and y be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)

Hypothesis

Ref	Expression
ax7w.1	⊢ (y = z → (φ ↔ ψ))

Assertion

Ref	Expression
ax7w	⊢ (∀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 ax7w

Step	Hyp	Ref	Expression
1		ax7w.1	. 2 ⊢ (y = z → (φ ↔ ψ))
2	1	alcomiw 1704	1 ⊢ (∀x∀yφ → ∀y∀xφ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540

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

This theorem is used by: (None)