Theorem 19.8aw 2050

Description: If a formula is true, then it is true for at least one instance. This is to 19.8a 2182 what spw 2033 is to sp 2184. (Contributed by SN, 26-Sep-2024.)

Hypothesis

Ref	Expression
19.8aw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
19.8aw	⊢ (𝜑 → ∃𝑥𝜑)

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

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

Proof of Theorem 19.8aw

Step	Hyp	Ref	Expression
1		alnex 1779	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		19.8aw.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	notbid 318	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
4	3	spw 2033	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ¬ 𝜑)
5	1, 4	sylbir 235	. 2 ⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
6	5	con4i 114	1 ⊢ (𝜑 → ∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1535  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778

This theorem is referenced by:  mo2icl  3736  eu6w  42631