Theorem ax11w 1994

Description: Weak version of ax-11 2021 from which we can prove any ax-11 2021 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2021, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)

Hypothesis

Ref	Expression
ax11w.1	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ax11w	⊢ (∀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 ax11w

Step	Hyp	Ref	Expression
1		ax11w.1	. 2 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
2	1	alcomiw 1958	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)