Theorem bnj1198 35425

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 1880	. 2 ⊢ (∃𝑥𝜓′ ↔ ∃𝑥𝜓)
4	1, 3	sylibr 237	1 ⊢ (𝜑 → ∃𝑥𝜓′)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  bnj1209  35426  bnj1275  35443  bnj1340  35453  bnj1345  35454  bnj605  35537  bnj607  35546  bnj906  35560  bnj908  35561  bnj1189  35639  bnj1450  35680  bnj1312  35688