Theorem lemma-L6.07a 772

Description: This lemma is used to construct an alternate definition of proper substitution.

'𝑥' cannot occur in '𝑡'.

Assertion

Ref	Expression
lemma-L6.07a	⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∃𝑥(𝑥 = 𝑡 ∧ 𝜑))

Distinct variable group:   𝑡,𝑥

Proof of Theorem lemma-L6.07a

Step	Hyp	Ref	Expression
1		lemma-L6.07a-L2 771	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∃𝑥(𝑥 = 𝑡 ∧ 𝜑))
2		lemma-L6.07a-L1 770	. 2 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))
3	1, 2	rcp-NDBII0 239	1 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∃𝑥(𝑥 = 𝑡 ∧ 𝜑))

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132  ∃wff-exists 595

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-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

This theorem is referenced by:  lemma-L6.08a  773