Theorem ax11wlem 1720

Description: Lemma for weak version of ax-11 1746. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax11w 1721. (Contributed by NM, 10-Apr-2017.)

Hypothesis

Ref	Expression
ax11wlemw.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
ax11wlem	⊢ (x = y → (φ → ∀x(x = y → φ)))

Distinct variable group:   ψ,x

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

Proof of Theorem ax11wlem

Step	Hyp	Ref	Expression
1		ax11wlemw.1	. 2 ⊢ (x = y → (φ ↔ ψ))
2		ax-17 1616	. 2 ⊢ (ψ → ∀xψ)
3	1, 2	ax11i 1647	1 ⊢ (x = y → (φ → ∀x(x = y → φ)))

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

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  ax11w  1721