Theorem bnj1198 35192

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 1877	. 2 ⊢ (∃𝑥𝜓′ ↔ ∃𝑥𝜓)
4	1, 3	sylibr 237	1 ⊢ (𝜑 → ∃𝑥𝜓′)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838

This proof depends on definitions:  df-bi 210  df-ex 1809

This theorem is used by:  bnj1209  35193  bnj1275  35210  bnj1340  35220  bnj1345  35221  bnj605  35304  bnj607  35313  bnj906  35327  bnj908  35328  bnj1189  35406  bnj1450  35447  bnj1312  35455