Theorem suballinf-P5 594

Description: Inference Version of '∀𝑥' Substitution Law.

For the deductive form with a variable restriction, see suballv-P5 623. For the most general form, see suball-P6 753.

Hypothesis

Ref	Expression
suballinf-P5.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
suballinf-P5	⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)

Proof of Theorem suballinf-P5

Step	Hyp	Ref	Expression
1		suballinf-P5.1	. . . . 5 ⊢ (𝜑 ↔ 𝜓)
2	1	rcp-NDBIEF0 240	. . . 4 ⊢ (𝜑 → 𝜓)
3	2	ax-GEN 15	. . 3 ⊢ ∀𝑥(𝜑 → 𝜓)
4	3	alloverim-P5.RC 589	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
5	1	rcp-NDBIER0 241	. . . 4 ⊢ (𝜓 → 𝜑)
6	5	ax-GEN 15	. . 3 ⊢ ∀𝑥(𝜓 → 𝜑)
7	6	alloverim-P5.RC 589	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜑)
8	4, 7	rcp-NDBII0 239	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)

Syntax hints:  ∀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

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by:  exnegall-P5  598  alloverimex-P5  601  lemma-L5.03a  666  example-E5.04a  675  genallw-P6  676  genexw-P6  677  gennallw-P6  678  gennexw-P6  679  nfrleq-P6  687  nfrneg-P6  688  solvesub-P6a  704  solvedsub-P6a  711  lemma-L6.02a  726  lemma-L6.03a  728  genex-P6  731  gennex-P6  738  lemma-L6.08a  773  psubleq-P6  783  psubnfr-P6  784  psubneg-P6  788  psuball2v-P6  796  psubex2v-P6  797