Theorem bnj1198 34771

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1198.1	⊢ (𝜑 → ∃𝑥𝜓)
bnj1198.2	⊢ (𝜓′ ↔ 𝜓)

Assertion

Ref	Expression
bnj1198	⊢ (𝜑 → ∃𝑥𝜓′)

Proof of Theorem bnj1198

Step	Hyp	Ref	Expression
1		bnj1198.1	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
2		bnj1198.2	. . 3 ⊢ (𝜓′ ↔ 𝜓)
3	2	exbii 1846	. 2 ⊢ (∃𝑥𝜓′ ↔ ∃𝑥𝜓)
4	1, 3	sylibr 234	1 ⊢ (𝜑 → ∃𝑥𝜓′)

Syntax hints:   → wi 4   ↔ wb 206  ∃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

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

This theorem is referenced by:  bnj1209  34772  bnj1275  34789  bnj1340  34799  bnj1345  34800  bnj605  34883  bnj607  34892  bnj906  34906  bnj908  34907  bnj1189  34985  bnj1450  35026  bnj1312  35034