Theorem suball-P7.RC 974

Description: Inference Form of suball-P7.RC 974. †

Hypothesis

Ref	Expression
suball-P7.RC.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
suball-P7.RC	⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)

Proof of Theorem suball-P7.RC

Step	Hyp	Ref	Expression
1		ndnfrv-P7.1 826	. . 3 ⊢ Ⅎ𝑥⊤
2		suball-P7.RC.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
4	1, 3	suball-P7 973	. 2 ⊢ (⊤ → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
5	4	ndtruee-P3.18 183	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)

Syntax hints:  ∀wff-forall 8   ↔ wff-bi 104  ⊤wff-true 153

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:  dfpsub-P7  978  suball-P7r.RC  1041  exnegall-P7  1046