Theorem alcomw 2042

Description: Weak version of alcom 2157 and biconditional form of alcomimw 2040. Uses only Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 28-Dec-2024.)

Hypotheses

Ref	Expression
alcomw.1	⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))
alcomw.2	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
alcomw	⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)

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

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

Proof of Theorem alcomw

Step	Hyp	Ref	Expression
1		alcomw.2	. . 3 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))
2	1	alcomimw 2040	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
3		alcomw.1	. . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))
4	3	alcomimw 2040	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
5	2, 4	impbii 209	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777

This theorem is referenced by:  cgsex4gOLD  3527  unissb  4944  dftr2c  5268  cotrg  6130  cnvsym  6135  dffun2  6573