Theorem dral1v 2386

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2460 with a disjoint variable condition, which does not require ax-13 2389. Remark: the corresponding versions for dral2 2459 and drex2 2463 are instances of albidv 1920 and exbidv 1921 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2177. (Revised by Wolf Lammen, 30-Mar-2024.)

Hypothesis

Ref	Expression
dral1v.1	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dral1v	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem dral1v

Step	Hyp	Ref	Expression
1		nfa1 2154	. . 3 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦
2		dral1v.1	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	albid 2223	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
4		axc11v 2264	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓))
5		axc11rv 2265	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜓))
6	4, 5	impbid 214	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜓))
7	3, 6	bitrd 281	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784

This theorem is referenced by:  drex1v  2387  drnf1v  2388