Theorem alcomimw 2050

Description: Weak version of ax-11 2168. See alcomw 2052 for the biconditional form. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Dec-2023.)

Hypothesis

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

Assertion

Ref	Expression
alcomimw	⊢ (∀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 alcomimw

Step	Hyp	Ref	Expression
1		alcomimw.1	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
2	1	cbvalvw 2043	. . . 4 ⊢ (∀𝑦𝜑 ↔ ∀𝑧𝜓)
3	2	biimpi 217	. . 3 ⊢ (∀𝑦𝜑 → ∀𝑧𝜓)
4	3	alimi 1818	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑧𝜓)
5		ax-5 1917	. 2 ⊢ (∀𝑥∀𝑧𝜓 → ∀𝑦∀𝑥∀𝑧𝜓)
6	1	biimprd 249	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝜓 → 𝜑))
7	6	equcoms 2027	. . . 4 ⊢ (𝑧 = 𝑦 → (𝜓 → 𝜑))
8	7	spimvw 1993	. . 3 ⊢ (∀𝑧𝜓 → 𝜑)
9	8	2alimi 1819	. 2 ⊢ (∀𝑦∀𝑥∀𝑧𝜓 → ∀𝑦∀𝑥𝜑)
10	4, 5, 9	3syl 18	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by:  excomimw  2051  alcomw  2052  hbalw  2058  ax11w  2141  bj-ssblem2  37002