Theorem cbvall-P7.VR12of2 1064

Description: cbvall-P7 1061 with two variable restrictions. †

'𝑥' cannot occur in '𝜓' and '𝑦' cannot occur in '𝜑'.

Hypothesis

Ref	Expression
cbvall-P7.VR12of2.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvall-P7.VR12of2	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦

Proof of Theorem cbvall-P7.VR12of2

Step	Hyp	Ref	Expression
1		ndnfrv-P7.1 826	. 2 ⊢ Ⅎ𝑥𝜓
2		ndnfrv-P7.1 826	. 2 ⊢ Ⅎ𝑦𝜑
3		cbvall-P7.VR12of2.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvall-P7 1061	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L11 28  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by:  nfreetjust-P8  1115