Theorem dral1vOLD 2362

Description: Obsolete version of dral1v 2361 as of 18-Nov-2024. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2167. (Revised by Wolf Lammen, 30-Mar-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
dral1vOLD	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem dral1vOLD

Step	Hyp	Ref	Expression
1		nfa1 2140	. . 3 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦
2		dral1v.1	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	albid 2210	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
4		axc11v 2250	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓))
5		axc11rv 2251	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜓))
6	4, 5	impbid 211	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜓))
7	3, 6	bitrd 278	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778

This theorem is referenced by: (None)